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The semiclassical limit on a star-graph with Kirchhoff conditions

The semiclassical limit on a star-graph with Kirchhoff conditions We consider the dynamics of a quantum particle of mass m on a n-edges star-graph −1 2 with Hamiltonian H =−(2m)   and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre˘ın’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators ⊕ ⊕ for the couple (H , H ), where H is the Hamiltonian with Dirichlet conditions in D D the vertex. Keywords Semiclassical dynamics · Quantum graphs · Coherent states · Scattering theory Mathematics Subject Classification 81Q20 · 81Q35 · 47A40 The authors acknowledge the support of the National Group of Mathematical Physics (GNFM-INdAM). B Claudio Cacciapuoti claudio.cacciapuoti@uninsubria.it Davide Fermi fermidavide@gmail.com https://fermidavide.com Andrea Posilicano andrea.posilicano@uninsubria.it DiSAT, Sezione di Matematica, Università dell’Insubria, via Valleggio 11, 22100 Como, Italy Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Present address: Dipartimento di Matematica ‘Guido Castelnuovo’, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Roma, Italy 0123456789().: V,-vol 45 Page 2 of 43 C. Cacciapuoti et al. 1 Introduction Aim of this work is to provide the semiclassical dynamics and scattering for an approx- imate coherent state propagating freely on a star-graph, in the presence of Kirchhoff conditions in the vertex. Since the pioneering work of Kottos and Smilansky [17], having in mind applica- tions to quantum chaos, the semiclassical limit of quantum graphs is often understood as the study of the distribution of eigenvalues (or resonances, see [18]) of self-adjoint −1 2 realizations of −(2m)   on the graph. To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2](seealso[3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs when (the reduced Planck constant) goes to zero. Inspired by the work of Hagedorn [12], instead, we look directly at the dynamics of the wave-function, for a class of initial states which are close to Gaussian coherent states supported on one of the edges of the graph. Closely related to our work is a series of papers by Chernyshev and Shafarevich [6, 8,9] in which the authors study the  → 0 limit of Gaussian wave packets propagating on graphs. Their main interest is the asymptotic growth (for large times) of the number of wave packets propagating on the graph. The main tool for the analysis is the complex WKB method by Maslov (see [19]). We also point out the work [7], by the same authors, in which they study the small  asymptotics of the eigenvalues of Schrödinger operators on quantum graphs (with Kirchhoff conditions in the vertices and in the presence of potential terms). In our previous work [5] we studied the semiclassical limit in the presence of a singular potential. Specifically, we considered the operator H , which is the quantum Hamiltonian in L (R) formally written as H =−  + α , where m is the mass α δ0 2m of the particle, is the Dirac-delta distribution centered in x = 0, and α is a real δ0 constant measuring the strength of the potential. Given a Gaussian coherent state on the real line of the form 1 1 i ψ (x ) := √ exp − (x − q) + p(x − q) for x ∈ R , σ,ξ 1/4 π σ 4σ σ (2 ) (1.1) with σ ∈ C,Re σ = σ > 0 and ξ ≡ (q, p) ∈ R , we studied the limit  → 0of −i H e . ,ξ To this aim we reasoned as follows. For fixed x ∈ R, consider the classical wave function defined by φ : R → C , φ (ξ) := ψ (x). σ,x σ,x σ,ξ Consider the vector field X (q, p) = (p/m, 0) associated with the free classical Hamiltonian h (q, p) = p /(2m) (q is the position and p the momentum of the 0 The semiclassical limit on a star-graph with Kirchhoff… Page 3 of 43 45 classical particle of mass m), and the Liouville operator p ∂ ∞ 2 dom(L) := C (R ), L := −iX ·∇ = − i . m ∂q Set pt p t it ξ := q + , p , A := , σ := σ + , t t 0 m 2m 2mσ where ξ is the solution of the free Hamilton equations, A is the (free) classical action, t t and σ takes into account the spreading of the wave function. If the dynamics is free (i.e., α = 0), one has the identity t i −i H  A 0 t e ψ (x ) = e ψ (x). σ ,ξ σ ,ξ 0 t The latter can be rewritten as t i −i H  A it L 0 t 0 e ψ (x ) = e e φ (ξ), (1.2) σ ,ξ ,x 0 t it L ∞ 2 2 2 where e is the realization in L (R ) of the strongly continuous (in L (R )) group of evolution generated by the self-adjoint operator L = L; explicitly, one it L has e f (ξ) = f (ξ ). ◦ ∞ Since H is a self-adjoint extension of H := H  C (R\{0}), mimicking the α 0 t i −i H  A it L α t β identity (1.2), we compared e ψ (x ) with e e φ (ξ), with L a σ ,x σ ,ξ 0 t ∞ 2 self-adjoint extension of L := L  C (M ), M := R \{(0, p) | p ∈ R}. Here, β 0 0 0 0 c is a real constant which parameterizes the self-adjoint extension, and it turns out that the optimal choice is β = 2 α / (see [5] for the details). In the same spirit, in the present work, we study the small  asymptotic of −i H e  where H is the quantum Hamiltonian defined as the self-adjoint real- σ ,ξ −1 2 ization of −(2m)   on the star-graph with Kirchhoff conditions in the vertex, and resembles a coherent state concentrated on one edge of the graph (see Sect. 1.3 σ,ξ below for the precise definition). The paper is structured as follows. In the remaining sections of the introduction we give the main definitions and results. Sections 2 and 3 contain a detailed description of the quantum and semiclassical dynamics on the star-graph respectively. In Sect. 4 we give the proofs of Theorems 1.4 and 1.6. Section 5 contains some additional remarks and comments. The paper is concluded by an “Appendix A” in which we present a proof of a technical result, namely an explicit formula for the wave operators for the pair (Dirichlet Laplacian, Neumann Laplacian) on the half-line. 1.1 Quantum dynamics on the star-graph By star-graph we mean a non-compact graph, with n edges (or leads) and one vertex. Each edge can be identified with a half-line, the origins of the half-lines coincide and identify the only vertex of the graph. 45 Page 4 of 43 C. Cacciapuoti et al. We recall that the Hilbert space associated with the star-graph is L (G) ≡ 2 2 ⊕ L (R ), with the natural scalar product and norm; in particular, for the L -norm =1 we use the notation 1/2 1/2 2 2 2 ≡ dx |ψ(x )| := dx | (x )| . L (G) G 0 =1 2 2 If ψ ∈ L (G), ψ ∈ L (R ) is its -th component with respect to the decomposi- n 2 tion ⊕ L (R ). In a similar way one can define the associated Sobolev spaces; in =1 2 n 2 particular, we set H (G) ≡⊕ H (R ), with the natural scalar product and norm. =1 We are primarily interested in the semiclassical limit of the quantum dynamics generated by the Kirchhoff Laplacian on the star-graph, which is the operator dom(H ) := ψ ∈ H (G) ψ (0) = ··· = ψ (0), ψ (0) = 0 , (1.3) K 1 n =1 H ψ := − ψ ; (1.4) 2m here ψ denotes the element of L (G) with components ψ , and ψ(0) (resp., ψ (0))the ψ ψ vector in C with components (0) (resp., (0)). Functions in dom(H ) are said to satisfy Kirchhoff (also called Neumann, or standard, or natural) boundary conditions. In the analysis of the semiclassical limit of the wave and scattering operators, we will have to fix a reference dynamics on the star-graph. To this aim we will consider the operator H [see also the equivalent definition in Eqs. (2.5)–(2.6) below] ⊕ 2 dom(H ) := ψ ∈ H (G) ψ (0) = ··· = ψ (0) = 0 , 1 n H ψ := − ψ ; 2m we remark that H can be understood as the direct sum of n copies of the Dirichlet Hamiltonian on the half-line (see Sect. 2.2 below for further details). We recall that the quantum wave operators and the corresponding scattering operator on L (G), are defined by t t ⊕ ± i H −i H := s-lim e e , (1.5) t →±∞ + ∗ − S := ( )  . (1.6) These operators can be computed explicitly (see Proposition 2.2 and Remark 2.3 below), and component-wisely for  = 1,..., n they read as follows: The semiclassical limit on a star-graph with Kirchhoff… Page 5 of 43 45 ± ∗ ψ = − (1 ∓ F F ) ψ , , s =1 where is the Kronecker delta, F and F are the Fourier-sine and Fourier-cosine , s c transforms respectively [see Eqs. (2.11) and (2.12)]; (S ψ) = − ψ . (1.7) =1 1.2 Semiclassical dynamics on a star-graph The generator of the semiclassical dynamics on the star-graph is obtained as a ∂ n 2 self-adjoint realization of the differential operator − i in ⊕ L (R × R), =1 m ∂q (q, p) ∈ R × R. To recover it we will make use of the method to classify the singular perturbations of self-adjoint operators developed by one of us in [20](seealso[21]). To do so, the first step is to identify a simple dynamics on the star-graph, more precisely its generator. We shall consider classical particles moving on the edges of the graph with elastic collision at the vertex. We start by considering the dynamics of a classical particle on the half-line with elastic collision at the origin. We obtain its generator as a limiting case from our previous work [5] and denote it by L . We postpone the precise definition of L to D D Sect. 3.1. Here we just note few facts. 2 2 L : dom(L ) ⊂ L (R × R) → L (R × R) is self-adjoint and acts on elements D D + + of its domain as p ∂ f (L f )(q, p) =− i (q, p) for (q, p) ∈ R × R . D + m ∂q For all t ∈ R, the action of the unitary evolution group associated with it is explicitly given by pt pt ⎨ f q + , p if q + > 0 , it L m m e f (q, p) = (1.8) pt pt − f − q − , −p if q + < 0 . m m The (trivial) classical dynamics of a particle on the star-graph with elastic collision at the vertex can be defined in the following way. Denote by f a function of the form ⎛ ⎞ f (q, p) ⎜ ⎟ n 2 . f ∈⊕ L (R × R), f (q, p) ≡ . . ⎝ ⎠ =1 f (q, p) n 45 Page 6 of 43 C. Cacciapuoti et al. If n = 1, | f (q, p)| dqdp can be interpreted as the probability of ⊕ L (R ×R) =1 finding a particle on the -th edge of the graph, with position in the interval [q, q +dq] and momentum in the interval [ p, p + dp]. ⊕ n Define the operator L := ⊕ L ; the associated dynamics is generated by the D =1 iL t n iL t unitary group e =⊕ e , and it is trivial in the sense that it can be fully =1 understood in terms of the dynamics on the half-line described above. We consider the map ( f ) (p) := lim f (q, p) for  = 1,..., n , q→0 defined on sufficiently smooth functions (we refer to Sect. 3 for the details). This map ⊕ ⊕ ⊕ can be extended to a continuous one on dom(L ). The operator L  ker( ) is D D symmetric; in Theorem 3.3, by using the approach developed in [20,21] we identify a family of self-adjoint extensions. Among those we select the one that turns out to be useful to study the semiclassical limit of exp(−iH t /) and denote it by L . K K We postpone the precise definition of L to Sect. 3.3, see, in particular, Remark 3.4. Here we just give component-wisely the formula for the associated unitary group, for  = 1,..., n and for all t ∈ R: pt pt ⎪ f q + , p if q + > 0 , m m it L e f (q, p) = 2 pt pt − f − q − , −p if q + < 0 . ⎪ , n m m =1 (1.9) We define the classical wave operators and the corresponding scattering operator n 2 on ⊕ L (R × R) by =1 ± it L −it L := lim e e (1.10) cl t →±∞ and + ∗ − S := ( )  . (1.11) cl cl cl These operators can be computed explicitly (see Proposition 3.8 below), and component-wisely they read as follows for  = 1,..., n ( is the Heaviside step function): f = − (∓p) f ; δ θ cl =1 (S f ) = − f . (1.12) cl  , =1 1.3 Truncated coherent states on the star-graph In general, there is no natural definition of a coherent state on a star-graph, neither there is a unique way to extend coherent states through the vertex. Since we are interested in The semiclassical limit on a star-graph with Kirchhoff… Page 7 of 43 45 initial states concentrated on one edge of the graph, we introduce the following class of initial states. We denote by ψ the unnormalized restriction of ψ [see Eq. (1.1)] σ,ξ σ ,ξ to R , namely, ψ ∈ L (R ), ψ (x ) = ψ (x ) for x > 0 . (1.13) σ,ξ σ,ξ σ,ξ On the graph we consider the quantum states defined as Definition 1.1 (Quantum states)Let σ ∈ C, with Re σ = σ > 0, and ξ = (q, p) ∈ R × R; consider any normalized function  ∈ L (R ), such that + + σ,ξ − ε | σ | − ψ ≤ C e for some C , ε > 0 . (1.14) 2 0 0 σ,ξ σ,ξ L (R ) We are primarily interested in quantum states on the star-graph of the form ⎛ ⎞ ,ξ ⎜ ⎟ ⎜ ⎟ ∈ L (G),  ≡ ⎜ ⎟ . σ,ξ σ,ξ ⎝ ⎠ Remark 1.2 In Definition 1.1 we assume that the constants C and ε do not depend on , ξ or σ. In what follows, whenever we refer to a state of the form  the constants σ,ξ C and ε are the ones given in Definition 1.1. One could choose the quantum states  (and the corresponding classical σ,ξ states given below) in a different way. As a matter of fact, for any choice of 2 2 2 2 ε σ ε σ the terms C exp(− q /( )) and C exp(− (q + pt /m) /(| | )) in the 0 0 t σ,ξ 0 bounds in Theorems 1.4 and 1.6 would be replaced by  − and σ ,ξ σ ,ξ 0 L (R ) − ψ respectively. σ ,ξ σ ,ξ t t L (R ) t t + Correspondingly, we will consider the family of classical states Definition 1.3 (Classical states) For any σ, ξ, and  as in Definition 1.1, consider σ,ξ the function  : R × R → C defined by σ,x (ξ) :=  (x). (1.15) σ,x σ,ξ We will make use of the family of classical states on the star-graph given by  ≡ ,x , 0,..., 0 . In general the functions  do not belong to L (R × R) but σ,x σ,x we will always assume that they are in L (R × R) (see Lemma 4.2, and Examples −it L 4.3 and 4.4 below). We remark that the classical operators e ,  , and S can be cl cl naturally extended to L (R × R). + 45 Page 8 of 43 C. Cacciapuoti et al. 1.4 Main results Our first result concerns the semiclassical limit of the dynamics. Theorem 1.4 Let σ > 0, ξ = (q, p) ∈ R × R and consider any initial state of the 0 + form  , together with its classical analogue  . Then, for all t ∈ R there holds σ ,ξ ,x 0 0 1/2 t i −i H  A it L K t K dx e  (x ) − e e  (ξ) σ σ ,x ,ξ 0 t 2 2 q 2 q (q+pt /m) − ε √ − − ε 2 2 σ 2 4σ | | 0 t 0 ≤ C e + 2C e + 2 e . (1.16) 0 0 Remark 1.5 Let t (ξ) := −mq/p be the classical collision time. Whenever |t − coll η η t |≤ m /| p| for some positive constant the second term on the right-hand coll 0 − ε η side of Eq. (1.16) is larger than 2C e . In the second part of our analysis we study the semiclassical limit of the wave operators and of the scattering operator. Theorem 1.6 Let σ > 0, ξ = (q, p) ∈ R × (R\{0}) and consider any state of the 0 + form  , together with its classical analogue  . Then, there hold σ ,x σ ,ξ 0 0 1/2 dx   (x ) − (  )(ξ) σ σ ,x ,ξ cl 0 0 2 2 q q 2 2 σ p √ − ε − 2 0 2 2 σ 4σ − 0 0 ≤ 2 C e + e + e , (1.17) and S  (x ) = (S  )(ξ). (1.18) cl σ σ ,x ,ξ 0 0 Identity (1.18) is an immediate consequence of Eqs. (1.7) and (1.12), and of the definitions of  and  . σ ,ξ σ ,x Remark 1.7 Equation (1.17) makes evident that   and (  )(ξ) are σ ξ cl σ , ,(·) 0 0 exponentially close (with respect to the natural topology of L (G)) in the semiclassical 2 2 2 2 + limit σ /q , /σ p → 0 for any ξ = (q, p) with q > 0 and p = 0. 0 0 As a matter of fact, it can be proved that the relation (1.17) remains valid also for p = 0 if one puts (  )(q, 0) =  (q, 0); the latter position appears to be σ ,x σ ,x cl 0 0 reasonable and is indeed compatible with the computations reported in the proof of 2 2 Proposition 3.8. Nonetheless, since exp(− σ p /) = 1 in this case, the resulting upper bound is of limited interest for what concerns the semi-classical limit. To say 2 2 more, for p = 0 and σ /q (or C ) small enough, by a variation of the arguments described in the proof of Theorem 1.6 one can derive the lower bound 1/2 ±  ± dx   (x ) − (  )(q, 0) σ ,(q,0) σ ,x cl 0 G The semiclassical limit on a star-graph with Kirchhoff… Page 9 of 43 45 2 2 q q − √ − ε 2 2 4σ σ 0 0 ≥ 1 − e − 2 C e . This shows that, as might be expected, the classical scattering theory does not provide a good approximation for the quantum analogue when p = 0. On the contrary, notice that Eq. (1.16) ensures a significant control of the error for the dynamics at any finite time t ∈ R even for p = 0. 2 The quantum theory 2.1 Dirichlet dynamics on the half-line Let us first consider the free quantum Hamiltonian for a quantum particle of mass m on the whole real line, defined as usual by 2 2 2 H : H (R) ⊂ L (R) → L (R), H ψ := − ψ , 0 0 2m 0 −i H together with the associated free unitary group U := e (t ∈ R). Correspond- ingly, let us recall that for any ∈ L (R) we have (x −y) m m 0 i 2 t (U ψ)(x ) = dy e ψ(y). (2.1) 2 π i t Let us further introduce the Dirichlet Hamiltonian on the half-line R , defined as usual by 1 2 dom(H ) := H (R ) ∩ H (R ), H ψ := − ψ , D + + D 2m D −i H and refer to the associated unitary group U := e (t ∈ R). As well known, the latter operator can be expressed as D − + U = U − U , (2.2) t t t where, in view of the identity (2.1), we introduced the bounded operators on L (R ) defined as follows for ψ ∈ L (R ) and x ∈ R : + + m m (x ±y) ± i 2 t (U ψ)(x ) := dy e ψ(y). (2.3) 2 i t Remark 2.1 Let us consider the bounded operator ψ(x ) if x > 0 , 2 2 : L (R ) → L (R), ( ψ)(x ) = 0if x < 0 , 45 Page 10 of 43 C. Cacciapuoti et al. together with its adjoint ∗ 2 2 ∗ : L (R) → L (R ), ( ψ)(x ) = ψ(x)(x ∈ R ). + + Namely,  gives the extension by zero to the whole real line R of any function on R , while  is the restriction to R of any function on R. Note that  is an isometry. In ∗ 2 ∗ fact,   is the identity on L (R ) and  is an orthogonal projector (but not the identity) on L (R); more precisely, we have ( is the Heaviside step function) ∗ 2 = 1 on L (R ), ∗ 2 2 ∗ : L (R) → L (R), (  ψ)(x ) = (x ) ψ(x)(x ∈ R). To proceed let us consider the parity operator 2 2 P : L (R) → L (R), (P ψ)(x ) = ψ(−x)(x ∈ R). Of course P is a unitary, self-adjoint involution which commutes with the free Hamil- tonian H , i.e., H P = PH . 0 0 Furthermore it can be checked by direct inspection that ran(P ) = ker( ) Using the bounded linear maps introduced above, one can express the operators defined in Eq. (2.3) as follows: − ∗ 0 + ∗ 0 ∗ 0 U =  U , U =  PU  =  U P . (2.4) t t t t t 0 ∗ 0 Recalling that (U ) = U , the above relation allow us to infer t −t − ∗ − + ∗ + (U ) = U ,(U ) = U . t −t t −t Let us finally point out that, on account of the obvious operator norms 1, = 1 and = 1, from Eq. (2.4) it readily follows ≤ 1 . 2.2 Dirichlet and Kirchhoff dynamics on the star-graph Let us now introduce the quantum Hamiltonian on the graph G, corresponding to Dirichlet boundary conditions at the vertex. This coincides with the direct sum of n copies of the Dirichlet Hamiltonian H on the half-line R , namely: D + ⊕ n n 1 2 dom(H ) := ⊕ dom(H ) = ⊕ H (R ) ∩ H (G), (2.5) D + D =1 =1 0 The semiclassical limit on a star-graph with Kirchhoff… Page 11 of 43 45 ⊕ n ⊕ 2 2 H := ⊕ H : dom(H ) ⊂ L (G) → L (G), D =1 D ⎛ ⎞ ⎜ . ⎟ H ψ := − . (2.6) ⎝ ⎠ D . 2m In view of the identity (2.2), it can be readily inferred that the corresponding unitary t ⊕ −i H group e (t ∈ R) can be expressed as t ⊕ −i H n − n + e =⊕ U −⊕ U , (2.7) =1 t =1 t where U is defined as in Eq. (2.3). To proceed let us consider the Kirchhoff Hamiltonian on the graph G. This is defined as in Eqs. (1.3)–(1.4). In what follows we denote by S the n×n matrix with components (S) := − , for ,  = 1,..., n . (2.8) , , By a slight abuse of notation we use the same symbol to denote the operator in L (G) defined by (S ψ) := (S) ψ ψ ∈ L (G). =1 By arguments similar to those given in the proof of [1, Thm. 2.1] (cf. also [11] and [15, Eq. (7.1)]) we get −i H n − n + e =⊕ U − S ⊕ U . (2.9) =1 t =1 t 2.3 The quantum wave operators and scattering operator Let us consider the wave operators and the corresponding scattering operator on L (G) respectively defined in Eqs. (1.5) and (1.6). Since H has purely absolutely continuous spectrum σ(H ) =[0, ∞),wehave K K ± 2 that  are unitary on the whole Hilbert space L (G), i.e., ± ∗ ± ( )  = 1 , which in turn ensures = 1 . (2.10) 1 ± Of course, the same identity (2.10) can be derived straightforwardly from the fact that  are defined as strong limits of unitary operators. 45 Page 12 of 43 C. Cacciapuoti et al. 2 2 2 Let us define the unitary operators F : L (R ) → L (R ) and F : L (R ) → s + + c + L (R ): 2i (F ψ)(k) := − √ dx sin(kx ) ψ(x)(k ∈ R ) ; (2.11) s + 2 π 0 (F ψ)(k) := √ dx cos(kx ) ψ(x)(k ∈ R ). (2.12) c + 2 π The wave operators can be computed explicitly. To this aim one could use the results from Weder [22] (see also references therein), with some modifications, since in [22] the reference dynamics is given by the Hamiltonian with Neumann boundary conditions. For the sake of completeness, we prefer to give an explicit derivation of the result, obtained by taking the limit t →±∞ on the unitary groups. We remark that in [22] the formulae are obtained by using the Jost functions. We have the following explicit formulae for the wave operators: Proposition 2.2 The quantum wave operators can be expressed as 1 1 ± n ∗ n ∗ = ⊕ (1 ± F F ) + S ⊕ (1 ∓ F F ). (2.13) s s =1 c =1 c 2 2 Proof By Eqs. (2.7) and (2.9) we easily obtain the identity ± n − − − + n + + + − = s-lim ⊕ (U U − U U ) + S ⊕ (U U − U U ) . (2.14) −t −t −t −t =1 t t =1 t t t →±∞ Let us find more convenient expressions for the operators on the right-hand side. Let ψ ∈ L (R ) and define ψ(x ) if x > 0 ψ(x ) if x > 0 ψ (x ) := ; ψ (x ) := . e o ψ(−x ) if x < 0 − ψ(−x ) if x < 0 ψ +ψ e o ∗ ∗ In view of Remark (2.1), we have:  ψ = ,  (ψ + ψ ) = 2 ψ, and  (ψ − e o e ψ ) = 0. Hence, see Eq. (2.4), − − ∗ 0 ∗ 0 U U ψ =  U  U  ψ −t t −t t ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) + U ( U ψ ) + U ( U ψ ) e e o −t t e −t t o −t t e 0 ∗ 0 +U ( U ψ ) −t t o (since U ψ is even and 0 ∗ 0 0 ∗ 0 U ψ is odd,( U ψ ) = U ψ ,( U ψ ) e o o e e o t t t t = U ψ , and we get) t o ∗ 0 ∗ 0 0 ∗ 0 =  ψ + U ( U ψ ) + U ( U ψ ) + ψ e o e e o o −t t −t t 4 The semiclassical limit on a star-graph with Kirchhoff… Page 13 of 43 45 1 1 ∗ 0 ∗ 0 0 ∗ 0 = ψ +  U ( U ψ ) + U ( U ψ ) . e o o e −t t −t t 2 4 On the other hand, − + ∗ 0 ∗ 0 ψ ψ U U =  U  U P −t t −t t ψ ψ ψ ψ ψ (Since P = P( + )/2 = ( − )/2 we get) e o e o ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) − U ( U ψ ) + U ( U ψ ) e e o −t t e −t t o −t t e 0 ∗ 0 −U ( U ψ ) −t t ∗ 0 ∗ 0 0 ∗ 0 =  ψ − U ( U ψ ) + U ( U ψ ) − ψ e o e o e o −t t −t t ∗ 0 ∗ 0 0 ∗ 0 ψ ψ =  −U ( U ) + U ( U ) . e o −t t o −t t e Hence 1 1 − − − + ∗ 0 ∗ 0 (U U − U U ) ψ = ψ +  U ( U ψ ) . o e −t t −t t −t t 2 2 Recall that U is the unitary group generated by the Dirichlet Laplacian on the half-line; its integral kernel is given by D 0 0 U (x , y) = U (x − y) − U (x + y) for x , y ∈ R . t t t Moreover, let U be the unitary group generated by the Neumann Laplacian on the half-line; its integral kernel is given by N 0 0 U (x , y) = U (x − y) + U (x + y) for x , y ∈ R . t t t Note that, for x ∈ R , ∗ 0 0 ψ ψ ( U )(x ) = dy U (x − y) (y) t o t o 0 0 D = dy U (x − y) − U (x + y) ψ(y) = (U ψ)(x), t t t similarly ∗ 0 0 0 N ψ ψ ψ ( U )(x ) = dy U (x − y) + U (x + y) (y) = (U )(x). t e t t t Hence, ∗ 0 ∗ 0 ∗ 0 D N D U ( U ψ ) =  U (U ψ) = U U ψ , o e e −t t −t t −t t 45 Page 14 of 43 C. Cacciapuoti et al. and 1 1 − − − + N D (U U − U U ) ψ = ψ + U U ψ . (2.15) −t −t t t −t t 2 2 A similar computation gives + + ∗ 0 ∗ 0 U U ψ =  U P  U P ψ −t t −t t ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) − U ( U ψ ) − U ( U ψ ) e e o e o e −t t −t t −t t 0 ∗ 0 +U ( U ) −t t o 1 1 ∗ 0 ∗ 0 0 ∗ 0 ψ ψ ψ = −  U ( U ) + U ( U ) , e o −t t o −t t e 2 4 and + − ∗ 0 ∗ 0 U U ψ =  U P  U  ψ −t t −t t ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 ψ ψ ψ =  U ( U ) + U ( U ) − U ( U ) e e o −t t e −t t o −t t e 0 ∗ 0 −U ( U ψ ) o o −t t ∗ 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) − U ( U ψ ) . o e e o −t t −t t Hence 1 1 1 1 + + + − ∗ 0 ∗ 0 N D (U U − U U ) ψ = ψ −  U ( U ψ ) = ψ − U U ψ . −t t −t t −t t o −t t 2 2 2 2 (2.16) N D To compute the wave operator we have to evaluate the limits s-lim U U ; t →±∞ −t t the latter give the wave operators  for the pair (Dirichlet Laplacian, Neumann ND Laplacian) on the half-line, which are computed in Proposition A.1 and equal ± F F . This, together with Eqs. (2.14), (2.15), and (2.16) concludes the proof of identity (2.13). Remark 2.3 Note that the wave operators do not depend on . Moreover the scattering operator is given by + ∗ − S = ( )  = S , (2.17) the same formula is written component-wisely in Eq. (1.7). The matrix S given here (v) (v) σ σ equals − , where is the scattering matrix at the vertex, as defined in [4, Def. 2.1.1.] (see also [4, Ex. 2.1.7, p. 41] and [16, Ex. 2.4]). The minus sign is due to the fact that as reference Hamiltonian we chose Dirichlet boundary conditions, instead of Neumann boundary conditions, see also [16]. The last identity in Eq. (2.17) can be derived by a simple computation starting from Eq. (2.13), recalling ∗ 2 that F and F are unitary operators, and noting that S = S, S = 1. s c The semiclassical limit on a star-graph with Kirchhoff… Page 15 of 43 45 3 The semiclassical theory 3.1 Classical dynamics on the half-line with elastic collision at the origin We start by recalling some basic definitions and results from [5]. Let X (q, p) = (p/m, 0) be the vector field associated with the free (classical) Hamiltonian of a particle of mass m and consider the differential operator 2 2 L : S (R ) → S (R ), Lf := −iX ·∇ f in the space of tempered distributions S (R ). We denote by p ∂ f 2 2 2 2 L : dom(L ) ⊆ L (R ) → L (R ), (L f )(q, p) =− i (q, p), 0 0 0 m ∂q 2 2 2 2 dom(L ) := f ∈ L (R ) Lf ∈ L (R ) , 2 0 −1 the maximal realization of L in L (R ). Posing R := (L − z) for z ∈ C\R one has im 0 imz(q−q )/p (R f )(q, p) = sgn(Imz) dq ((q − q ) pImz) e f (q , p). (3.1) | p| it L Moreover the action of the (free) unitary group e (t ∈ R)isgiven by pt it L e f (q, p) = f q + , p . For any f ∈ S(R ) we define the map ( f )(p) := f (0, p). For a comparison with the results in [5, Sec. 2], recall that the map can be equivalently defined as ˜ ˜ γ √ ( f )(p) = dk f (k, p) where f (k, p) is the Fourier transform of f (q, p) 2 π γ γ in the variable q.By[5, Lem. 2.1], the map extends to a bounded operator : 2 2 2 dom(L ) → L (R, | p| dp), where dom(L ) ⊂ L (R ) is endowed with the graph 0 0 0 2 2 2 norm. Hence, for any z ∈ C\R the operator R : L (R ) → L (R, | p| dp) is bounded, and so is its adjoint (in z ¯): 2 −1 2 2 0 ∗ G : L (R, | p| dp) → L (R ), G := ( R ) z z z ¯ 2 −1 2 (here L (R, | p| dp) and L (R, | p|dp) are considered as a dual couple with respect to the duality induced by the scalar product in L (R)). An explicit calculation gives im imzq/p (G u)(q, p) = (qpImz) sgn(Imz) e u(p). z θ | p| Next we consider the classical motion of a point particle of mass m on the whole real line, with elastic collision at the origin. The generator of the dynamics, denoted 45 Page 16 of 43 C. Cacciapuoti et al. by L , is obtained as a limiting case, for β →∞, of the operator L defined in [5]. ∞ β To this aim we set 2| p| ∞ 2 2 −1 ∞ : L (R, | p|dp) → L (R, | p| dp), ( u)(p) := isgn(Imz) u(p). z z In addition, let us consider the projector on even functions (here either (p) =| p| or −1 (p) =| p| ) 2 2 ρ ρ : L (R, dp) → L (R, dp), ( f )(p) := f (p) + f (−p) . ev ev By [5, Thm. 2.2], here employed with β →∞, the operator L is defined by 2 2 ∞ dom(L ) := f ∈ L (R ) f = f + G f , f ∈ dom(L ) , ∞ z z ev z z 0 (L − z) f = (L − z) f , ∞ 0 z ∞ −1 for all z ∈ C\R. The associated resolvent operator R := (L − z) (z ∈ C\R) can be expressed as follows, in terms of the free resolvent R and of the trace operator # $ ∞ 0 imzq/p 0 0 γ γ (R f )(q, p) = (R f )(q, p) − (qpImz) e ( R f )(p) + ( R f )(−p) . z z z z (3.2) More explicitly, we have im ∞ imz(q−q )/p (R f )(q, p) = sgn(Imz) dq (q − q ) pImz e f (q , p) | p| imz(q−q )/p − (qpImz) (−q pImz) e f (q , p) θ θ imz(q+q )/p + (q pImz) e f (q , −p) . (3.3) Correspondingly, let us recall that [5, Prop. 2.4] gives, for all t ∈ R, it L it L ∞ 0 e f (q, p) = e f (q, p) (3.4) |pt | it L it L 0 0 − θ(−tqp) θ −|q| e f (q, p) + e f (−q, −p) it L it L 0 0 (here, for a comparison with [5], we used e f (−· , −· ) (q, p) = e f (−q, −p) ). To proceed, let us introduce the lateral traces defined by # $ ∞ 0 0 0 γ γ γ γ (R f ) (p) := ( R f )(p) − θ(pImz) ( R f )(p) + ( R f )(−p) z z z z 0 0 γ γ = (−pImz)( R f )(p) − (pImz)( R f )(−p), (3.5) θ θ z z # $ ∞ 0 0 0 γ γ γ γ (R f ) (p) := ( R f )(p) − (−pImz) ( R f )(p) + ( R f )(−p) − θ z z z z 0 0 γ γ = (pImz)( R f )(p) − (−pImz)( R f )(−p) (3.6) θ θ z z The semiclassical limit on a star-graph with Kirchhoff… Page 17 of 43 45 (here we used the trivial identity (−s) = 1 − (s) ). Clearly, (R f ) are odd θ θ functions and, using again [5,Lem.2.1], : dom(L ) → L (R, | p|dp) (3.7) ± ∞ odd is a bounded operator. We remark that the action of can be understood as ∞ ∞ (R f ) (p) = lim ± (R f )(q, p). ± q→0 z z For the subsequent developments it is convenient to express the free resolvent R in terms of R . More precisely, we have the following explicit characterization. Lemma 3.1 For any z ∈ C\R and for any f ∈ L (R ), there holds 0 ∞ (R f )(q, p) = (R f )(q, p) z z imzq/p ∞ ∞ γ γ − θ(qpImz) sgn(q) e ( R f )(p) − ( R f )(p) . + − z z Proof From Eqs. (3.5) and (3.6) we readily infer that ∞ 0 γ γ (−pImz)( R f )(p) = (−pImz)( R f )(p), θ θ z z ∞ 0 γ γ (pImz)( R f )(p) = (pImz)( R f )(p). θ − θ z z The above relations imply, in turn, 0 ∞ ∞ γ γ γ ( R f )(p) = (−pImz)( R f )(p) + (pImz)( R f )(p), θ + θ − z z z and, since R f are odd functions, 0 ∞ ∞ γ γ γ ( R f )(−p) =− θ(pImz) R f (p) − θ(−pImz) R f (p). + − z z z Substituting the latter identities into Eq. (3.2) and noting that (qpImz) = (q) + (−q) (qpImz) = (q) (pImz) + (−q) (−pImz), θ θ θ θ θ θ θ θ and sgn(q) (qpImz) = (q) (pImz) − (−q) (−pImz), θ θ θ θ θ we obtain (R f )(q, p) 0 imzq/p ∞ = (R f )(q, p) − e (q) (pImz)( R f )(p) θ θ − z z − (q) (pImz)( R f )(p) θ θ + ∞ ∞ γ γ + (−q) (−pImz)( R f )(p) − (−q) (−pImz)( R f )(p) θ θ + θ θ − z z 45 Page 18 of 43 C. Cacciapuoti et al. 0 imzq/p ∞ ∞ γ γ = (R f )(q, p) + (qpImz) sgn(q) e ( R f )(p) − ( R f )(p) , θ + − z z z which suffices to infer the thesis. Similarly, for the unitary operator describing the dynamics we have 2 2 Lemma 3.2 For any t ∈ R and for any f ∈ L (R ), there holds it L it L e f (q, p) = e f (q, p) |pt | it L it L ∞ ∞ − (−tqp) −|q| e f (q, p)+ e f (−q,−p) . θ θ Proof First note that the identity in Eq. (3.4) entails it L it L ∞ 0 e f (−q, −p) = e f (−q, −p) |pt | it L it L 0 0 − θ(−tqp) θ −|q| e f (q, p)+ e f (−q,−p) . From the above relation and the previously cited equation, we derive it L it L ∞ ∞ e f (q, p) + e f (−q, −p) |pt | it L it L 0 0 = 1 − 2 θ(−tqp) θ −|q| e f (q, p) + e f (−q, −p) , which in turn implies |pt | it L it L ∞ ∞ (−tqp) −|q| e f (q, p) + e f (−q, −p) θ θ |pt | it L it L 0 0 =− (−tqp) −|q| e f (q, p) + e f (−q, −p) . θ θ The thesis follows upon substitution of the above identity into Eq. (3.4). Let us now consider the natural decomposition 2 2 2 L (R ) ≡ L (R ∪ R )×R, dqdp − + 2 2 2 = L (R , dq) ⊕ L (R , dq) ⊗ L (R, dp) − + 2 2 2 2 = L (R , dq) ⊗ L (R, dp) ⊕ L (R , dq) ⊗ L (R, dp) − + 2 2 = L (R × R) ⊕ L (R × R), (3.8) − + All the equalities in Eq. (3.8) must be understood as isomorphisms of Hilbert spaces (see, e.g., [13,p. 85]). The semiclassical limit on a star-graph with Kirchhoff… Page 19 of 43 45 2 2 2 and notice that both the subspaces L (R ×R) ≡ L (R ×R)⊕{0} and L (R ×R) ≡ − − + 2 ∞ {0}⊕ L (R × R) are left invariant by the resolvent R , this is evident from Eq. (3.3). Taking this into account, we introduce the bounded operator D 2 2 D ∞ R : L (R × R) → L (R × R), (R f )(q, p) := R (0 ⊕ f ) (q, p). (3.9) + + z z z By direct computations, from Eq. (3.3) (here employed with q > 0) we get R f (q, p) im imz(q−q )/p = sgn(Imz) dq θ (q − q )pImz e f (q , p) | p| imz(q+q )/p − (pImz) e f (q , −p) im imzq/p −imzq /p = sgn(Imz) e (pImz) dq e f (q , p) | p| imzq /p − dq e f (q , −p) −imzq /p + θ(−pImz) dq e f (q , p) . 2 D We denote by L the self-adjoint operator in L (R × R) having R as resolvent, D + so that dom(L ) = f ∈ L (R × R) (0 ⊕ f ) ∈ dom(L ) , L f := L (0 ⊕ f ). D + ∞ D ∞ (3.10) For all (q, p) ∈ R ×R, t ∈ R and f ∈ L (R ×R), from the above definition and from + + Eq. (3.4) we get it L it L D ∞ e f (q, p) = e (0 ⊕ f ) (q, p) pt it L = q + e (0 ⊕ f ) (q, p) pt it L − − q − e (0 ⊕ f ) (−q, −p), (3.11) which describes the motion of a classical particle on the half-line R with elastic collision at q = 0. Let us also mention that, in view of the basic identity pt it L it L 0 0 e (0 ⊕ f ) (q, p) = θ q + e (0 ⊕ f ) (q, p), (3.12) |pt | pt pt Note that for q > 0we have θ(−tqp) θ −|q| = θ(−tp) θ − − q = θ −q − = m m m pt 1 − q + . m 45 Page 20 of 43 C. Cacciapuoti et al. the above relation (3.11) is equivalent to it L it L it L D 0 0 e f (q, p) = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, −p). (3.13) Another equivalent (and more explicit) formula for the action of the unitary group it L e is the one given in Eq. (1.8). Finally, from Lemmata 3.1 and 3.2 (here employed with q > 0) we derive, respec- tively, 0 D imzq/p D R (0 ⊕ f ) (q, p) = (R f )(q, p) − (pImz) e ( R f )(p), θ + z z z pt it L it L 0 D e (0 ⊕ f ) (q, p) = θ q + e f (q, p) pt it L − −q − e f (−q, −p). (3.14) 3.2 Classical dynamics on the graph with total reflection at the vertex n 2 Let us now consider the “classical” Hilbert space ⊕ L (R × R) and indicate any =1 of its elements with the vector notation ⎛ ⎞ f (q, p) ⎜ ⎟ n 2 f ∈⊕ L (R × R), f (q, p) ≡ . + ⎝ ⎠ =1 . f (q, p) Let L be defined according to Eq. (3.10), and consider the classical dynamics on the star-graph G with total elastic collision in the vertex; this is described by the self-adjoint operator ⊕ n L := ⊕ L . D =1 The associated resolvent and time evolution operators are respectively given by D⊕ −1 n D R := (L − z) =⊕ R , z z D =1 it L n it L e =⊕ e . =1 Explicitly, for  = 1,..., n and t ∈ R,fromEq. (1.8) we derive pt pt f q + , p if q + > 0 , m m it L e f (q, p) = pt pt − f − q − , −p if q + < 0 . m m The semiclassical limit on a star-graph with Kirchhoff… Page 21 of 43 45 3.3 Singular perturbations of the classical dynamics on the graph Let us consider the restriction to dom(L ) of the trace map introduced in Eq. (3.7); D + this defines a bounded operator : dom(L ) → L (R, | p|dp). + D odd We use the above map to define a trace operator on the graph: n n n 2 γ γ := ⊕ :⊕ dom(L ) →⊕ L (R, | p|dp). + D =1 =1 =1 odd In what follows we use the technique developed by one of us in [20,21] to char- ⊕ ⊕ acterize all the self-adjoint extensions of the symmetric operator L  ker( ) (see Theorem 3.3 below). Among those we select the one that turns out to be useful to study the semiclassical limit of exp(−iH t /),see Remark 3.4. To proceed, we introduce the operator ⊕ ⊕ D⊕ n + n 2 n 2 ˘ ˘ G := R ≡⊕ G :⊕ L (R × R) →⊕ L (R, | p|dp), z + z =1 z  =1 odd + D G := R , z z and its adjoint with complex conjugate parameter, ⊕ ⊕ n + n 2 n 2 G := G ≡⊕ G :⊕ L (R, | p|dp) →⊕ L (R × R), z =1 z  odd =1 z ¯ + + G := G . z ¯ Note that by Eq. (3.1) one has im 0 −imzq /p R (0 ⊕ f ) (p) = sgn(Imz) dq θ(−pImz) e f (q , p). | p| From the latter identity, together with Eqs. (3.9) and (3.5), we derive + ∞ (G f )(p) = R (0 ⊕ f ) (p) z z 0 0 γ γ = (−pImz) R (0 ⊕ f ) (p) − (pImz) R (0 ⊕ f ) (−p), θ θ z z im −imzq /p =sgn(Imz) dq (−pImz) e f (q , p) | p| imzq /p − (pImz) e f (q , −p) . (3.15) 2 2 In view of the latter expression, for all f ∈ L (R × R) and any φ ∈ L (R, | p|dp) odd we have + + φ φ dqdp (G )(q, p) f (q, p) = dp (p)(G f )(p) z ¯ R ×R R + 45 Page 22 of 43 C. Cacciapuoti et al. im −imzq ¯ /p = dp φ(p) sgn(Imz ¯) dq (−pImz ¯) e f (q, p) | p| R 0 imzq ¯ /p − (pImz ¯) e f (q, −p) im −imzq ¯ /p φ φ = dqdp (p) − (−p) θ(pImz) sgn(Imz) − e f (q, p) | p| R ×R 2im imzq/p = dqdp (pImz) sgn(Imz) e φ(p) f (q, p), | p| R ×R which proves that im + imzq/p (G φ)(q, p) = 2 g (q, p) φ(p), g (q, p) := (pImz) sgn(Imz) e . z z | p| (3.16) On account of the identities im (g − g ) (p) = sgn(Imz) − sgn(Imw) , + z w 2 | p| which can be easily checked by a direct calculation (see also [5,p.7]),and + + + ∗ + (G − G ) = (z − w) (G ) G , z w z w ¯ which is consequence of the first resolvent identity (see [20, Lem. 2.1], paying attention to the different sign convention in the definition of the resolvent), we have that the linear map + 2 −1 2 dom(M ) := L (R, | p| dp) ∩ L (R, | p|dp), z odd odd + + 2 −1 2 M : dom(M ) ⊂ L (R, | p| dp) → L (R, | p|dp), z z odd odd im + ∞ ∞ φ φ (M )(p) := 2 m (p) (p), m (p) := − sgn(Imz) , (3.17) z z z 2| p| satisfies the identities + ∗ + + + + ∗ + (M ) = M , M − M = (w − z)(G ) G . z z w z z ¯ w ¯ Hence, setting ⊕ n + n 2 −1 n 2 M :⊕ dom(M ) ⊂⊕ L (R, | p| dp) →⊕ L (R, | p|dp), z =1 z =1 odd =1 odd ⊕ n + M := ⊕ M , z =1 z one gets the identities ⊕ ∗ ⊕ ⊕ ⊕ ⊕ ∗ ⊕ (M ) = M , M − M = (w − z)(G ) G . z z ¯ z w w ¯ z The semiclassical limit on a star-graph with Kirchhoff… Page 23 of 43 45 n n To proceed, let us consider any orthogonal projector : C → C and any self-adjoint n n operator B : C → C , represented by the matrices with components ( ) and (B) respectively. By a slight abuse of notation we use the same symbols to denote the corresponding operators on vector valued functions; e.g., for f =⊕ f ∈ =1 n 2 n 2 ⊕ L (R × R), one has f ∈⊕ L (R × R) with components ( = 1,..., n) + + =1 =1 f ) = ( ) f , =1 n n 2 n n or, for ⊕ φ ∈⊕ L (R, dp), one has (⊕ φ ) := ⊕ =1 =1 odd =1 =1 ) , and similarly for B. Then, by [20, Thm. 2.1] here employed j =1 with τ := , we obtain the following ⊕ ⊕ Theorem 3.3 Let z ∈ C\R. Assume that M = M and B = B . Then, the z z linear bounded operator −1 ,B D⊕ ⊕ ⊕ ⊕ R := R + G B + M G (3.18) z z z z z is the resolvent of a self-adjoint extension L of the densely defined, closed sym- ,B ⊕ ⊕ metric operator L  ker( ). Such an extension acts on its domain n 2 dom(L ) := f ∈⊕ L (R × R) ,B =1 −1 ⊕ ⊕ ⊕ ⊕ f = f + G B + M f , f ∈dom L z z z z z + D according to L − z f = L − z f . (3.19) ,B Remark 3.4 We use the notation L (where K stands for Kirchhoff) to denote the self-adjoint extension corresponding to the choices ⎛ ⎞ 11 ··· 1 ⎜ ⎟ 11 ··· 1 ⎜ ⎟ B = 0, = . (3.20) ⎜ . . .⎟ . . . . n ⎝ ⎠ . . . 11 ··· 1 K −1 We denote the associated resolvent operator with R := (L − z) . In the sequel, we proceed to determine the unitary evolution associated with the above choices by means of arguments analogous to those described in the proof of [5, Prop. 2.4]. 45 Page 24 of 43 C. Cacciapuoti et al. Proposition 3.5 For all f ∈⊕ L (R × R) and for all t ∈ R there holds =1 ⎛   ⎞ ⎛   ⎞ it L it L 0 0 e (0⊕ f ) (q, p) e (0⊕ f ) (−q, −p) 1 1 ⎜ . ⎟ ⎜ . ⎟ it L . . (e f )(q, p) = − S (3.21) ⎝ ⎠ ⎝ ⎠ . . it L it L 0 0 e (0⊕ f ) (q, p) e (0⊕ f ) (−q, −p) n n where S := 1 − 2 was already defined in relation with the quantum scattering operator, see Eq. (2.8). Proof Throughout the whole proof we work component-wisely, denoting with  ∈ {1, ..., n} a fixed index. Let us first remark that the resolvent (3.18), with B, as in Eq. (3.20), acts on any element f ∈⊕ L (R × R) according to =1 n n K D D R f (q, p) = (R f )(q, p) + 2 g (q, p) ( ) ( R f )(p) z j j + j z z z 2 m (p) j =1 j =1 g (q, p) 1 D D = (R f )(q, p) + ( R f )(p). + j z z m (p) n j =1 From the above relation we derive the following, recalling the explicit expressions for g (q, p) and m (p) given in Eqs. (3.16) and (3.17), as well as Eq. (3.15)for D + R f = G f : z z K D R f (q, p) = (R f )(q, p) z z 2im imz(q+q )/p + (pImz) sgn(Imz) dq e f (q , −p). n | p| j =1 −it L We now proceed to compute the unitary operator e (t ∈ R) by inverse Laplace transform, using the above representation for the resolvent R . Let us first assume t > 0; then, for any c > 0 and f ∈ dom(L ), we get (see [10, Ch. III, Cor. 5.15]) r +ic −it L −izt K e f = lim dz e R f r →∞ 2 π i −r +ic ct r −itk K = lim dk e R f . k+ic π r →∞ 2 i −r On the one hand, recalling Eq. (3.13)wehave ct r −itk D −it L lim dk e (R f )(q, p) = (e f )(q, p) k+ic r →∞ 2 π i −r −it L −it L 0 0 = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, −p). The semiclassical limit on a star-graph with Kirchhoff… Page 25 of 43 45 On the other hand, noting that pIm(k+ic) = (pc) = (p) and sgn Im(k+ic) = θ θ θ sgn(c) =+1for c > 0, by computations similar to those reported in the proof of [5, Prop. 2.4] we get ct r e 2im −itk lim dk e pIm(k +ic) sgn Im(k +ic) r →∞ 2 π i n| p| −r im(k+ic)(q+q )/p dq e f (q , −p) j =1 c(t −mq/p) r 2m e i (mq/p−t )k = (p) lim dk e r →∞ n| p| 2 π −r j =1 im(k+ic)q /p dq e (q ) f (q , −p) θ j 2m c(t −mq/p) = (p) e n| p| −1 F F ( · ) f ( · , −p) − m(∗+ ic)/p (mq/p − t ) j =1 2 pt pt = (p) − q + f −q + , −p θ θ n m m j =1 −it L = e (0 ⊕ f ) (−q, −p). j =1 Summing up, the above relations imply −it L −it L −it L K 0 0 e f = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, −p) −it L + e (0 ⊕ f ) (−q, −p). (3.22) j =1 For t < 0 one can perform similar computations, starting from the following identity where c > 0: r −ic −it L −izt K e f =− lim dz e R f π r →∞ 2 i −r −ic −ct r −itk K =− lim dk e R f . k−ic r →∞ 2 π i −r Especially, recall the following basic identity regarding the unitary Fourier transform F and its inverse −1 F : cq −1 F Fh a(∗+ ic) (q) = h(q/a) fora ∈ R\{0}, c > 0, q ∈ R , |a| c· e 2 which holds true whenever h(·/a) ∈ L (R). In addition, keep in mind the relation written in Eq. (3.12). |a| 45 Page 26 of 43 C. Cacciapuoti et al. We omit the related details for brevity. In the end, one obtains exactly the same expres- sion as in Eq. (3.22), which with the trivial replacement t →−t proves Eq. (3.21). it L Remark 3.6 By Eq. (3.12) we infer that the action of the unitary group e is explicitly given, component-wisely, by Eq. (1.9). Remark 3.7 Recalling the explicit form of (see Eq. (3.20)) we obtain ⎛ ⎞ ⎛ ⎞ n − 2 −2 ··· −2 n − 1 −1 ··· −1 ⎜ ⎟ ⎜ ⎟ −2 n − 2 ··· −2 −1 n − 1 ··· −1 1 1 ⎜ ⎟ ⎜ ⎟ S = , 1 − = . (3.23) ⎜ ⎟ ⎜ ⎟ . . . . . . . . . . . . . . . . ⎝ ⎠ ⎝ ⎠ n n . . . . . . . . −2 −2 ··· n − 2 −1 −1 ··· n − 1 In particular, for a star-graph with three edges (n = 3) we have ⎛ ⎞ 1 −2 −2 ⎝ ⎠ S = −21 −2 , −2 −21 whence S =− M with respect to the notation used in [1]. 3.4 The semiclassical wave operators and scattering operator Let us consider the wave operators and the corresponding scattering operator on ⊕ L (R ×R), respectively defined by Eqs. (1.10) and (1.11). The following propo- =1 sition provides explicit expressions for these operators. Proposition 3.8 The limits in Eq. (1.10) exist point-wisely for any ξ ≡ (q, p) ∈ R × 2 n 2 (R\{0}) and in L (R × R) for any f ∈⊕ L (R × R); moreover, there holds + + =1 # $ # $ f (q, p) = 1 − (∓p) 2 f (q, p) = (±p) 1 + (∓p) S f (q, p). θ θ θ cl (3.24) Furthermore, the scattering operator is given by S = 1 − 2 = S . (3.25) cl Proof First of all let us point out that for all t ∈ R and any  ∈{1, ..., n},Eqs.(3.13) and (3.21) give, respectively, it L it L it L D 0 0 e f (q, p) = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, − p) pt pt = (0 ⊕ f ) q + , p − (0 ⊕ f ) −q − , − p m m pt pt pt pt = θ q + f q + , p − θ −q − f −q − , − p ; m m m m −it L −it L −it L K 0 0 (e f ) (q, p) = e (0 ⊕ f ) (q, p) + 2 −1 e (0 ⊕ f ) (−q, − p) j =1 The semiclassical limit on a star-graph with Kirchhoff… Page 27 of 43 45 pt pt = (0 ⊕ f ) q − , p + 2 −1 (0 ⊕ f ) −q + , − p m m j =1 pt pt = q − f q − , p m m pt pt + 2 −1 θ −q + f −q + , − p . m m j =1 In view of the above relations, recalling that q > 0, by direct computations we obtain it L −it L it L −it L K D K e e f (q, p) = e (e f ) (q, p) pt pt −it L = q + (e f ) q + , p m m pt pt −it L − −q − (e f ) −q − , −p m m pt pt = q + f (q, p) − −q − 2 −1 f (q, p) θ θ m m j =1 pt = f (q, p) − θ −q − 2 f (q, p), j =1 which gives ⊕ pt it L −it L e e f (q, p) = f (q, p) − −q − 2 f (q, p). pt Then Eq. (3.24) follows noting that −q − → (∓p) for t →±∞. θ θ Next, since is symmetric, by elementary arguments we get # $ ± ∗ ( ) f (q, p) = 1 − (∓p) 2 f (q, p). cl On account of the above identity we obtain # $ + ∗ − − S f (q, p) = ( )  f (q, p) = 1 − θ(−p) 2 f (q, p) cl cl cl cl # $# $ # $ = 1 − θ(−p) 2 1 − θ(p) 2 f (q, p) = 1 − 2 f (q, p), which proves Eq. (3.25). 4 Comparison of the semiclassical and quantum theories Recall the definition of coherent states in L (R) given in Eqs. (1.1), and (1.2) describ- ing their free evolution. 45 Page 28 of 43 C. Cacciapuoti et al. −it L Additionally, notice that the action of the operators e ,  , and S can be cl cl extended in a natural way to ⊕ L (R × R). =1 For later reference let us point out the following auxiliary result, regarding the functions ψ defined in Eq. (1.13), and the operators U defined in Eq. (2.3). σ,ξ t Lemma 4.1 For all ξ = (q, p) ∈ R × R and for any t ∈ R there holds i 1 ± A 4σ U ψ − e ψ ≤ √ e . σ ξ σ ξ 2 t , ,∓ 0 t t L (R ) Proof Let us first remark that, on account of the considerations reported in Remark 2.1, ψ =  ψ and for any x > 0 we have the chain of identities σ ,ξ σ ,ξ 0 0 ± 0  0 U ψ (x ) = U ψ (∓x ) = U ψ (∓x ) + E (∓x ) t σ ,ξ t σ ,ξ t σ ,ξ t 0 0 0 = e (∓x ) + E (∓x ) σ ξ t t t i i A   A t t = e ψ (x ) + E (∓x ) = e ψ (x ) + E (∓x), σ ,∓ξ t σ ,∓ξ t t t t where we put E (x ) := − U (1 − ) ψ (x ) for convenience of notation and used t t σ ,ξ the trivial identity ψ (−x ) = ψ (x ). σ,ξ σ,− ξ On the other hand, via an explicit calculation involving the trivial inequality 2 2 2 η η − (a+b) − (a +b ) e ≤ e for , a, b ≥ 0 we obtain 2 2 2 2 E (∓· ) ≤ E = U (1 − ) ψ = (1 − ) ψ 2 2 θ 2 θ 2 t t t σ ,ξ σ ,ξ L (R ) L (R) 0 L (R) 0 L (R) 2 2 (x +q) q 0 ∞ − − 1 2 1 2 2 2σ 2σ 0 0 = dx |ψ (x )| = √ dx e ≤ e , (4.1) σ ,ξ 2 π  σ −∞ 0 which proves the thesis in view of the previous arguments. To proceed let us point out the forthcoming lemma which characterizes a large class of functions satisfying the condition in Definition 1.1. Lemma 4.2 Let > 0, ξ = (q, p) ∈ R × R and consider a family of functions ∞ ∞ χ η ∈ L (R ), uniformly bounded in L (R ) with respect to , q and such that q, + + η(x ) = 1 for |x − q| < q . (4.2) q, Then, the functions  ∈ L (R ) defined by σ,ξ (x ) := (x ) ψ (x)(x > 0) (4.3) q,η σ,ξ ,ξ q, σ,ξ L (R ) fulfill the condition (1.14) with ε < min{1/4, /8} and C depending only on ∞ . q, L (R ) + The semiclassical limit on a star-graph with Kirchhoff… Page 29 of 43 45 Proof Let us first remark that the states  defined in Eq. (4.3) have unit norm in ,ξ L (R ) by construction; moreover, again from Eq. (4.3) it follows that ψ = + q, σ,ξ . Taking these facts into account, we have η 2 q, σ,ξ L (R ) σ + ,ξ χ χ ψ ψ ψ − ≤  − η + ( η − 1) 2 q, 2 q, 2 σ ξ σ,ξ σ ξ σ,ξ σ,ξ , L (R ) , L (R ) L (R ) + + + = 1 − q, 2 σ,ξ L (R ) σ,ξ + L (R ) + ( η − 1) ψ q, 2 σ,ξ L (R ) χ χ = 1 − + ( − 1) ψ . η 2 η q, q, 2 σ,ξ L (R ) σ,ξ + L (R ) On one hand, recalling the definition (1.13)of ψ and that = 1, using σ,ξ σ ξ L (R) 2 2 2 the basic inequality (a − b) ≤|a − b | for a, b > 0 we get 2 2 χ χ 1 − η 2 η q, 2 q, 2 σ,ξ L (R ) σ,ξ σ,ξ + L (R) L (R ) = dx |ψ (x )| − dx (x ) ψ (x ) q, σ,ξ σ,ξ R 0 2 2 ≤ dx 1 − (x ) (x ) ψ (x ) θ η q, σ,ξ 2 2 2 ≤ dx 1 − (x ) ψ (x ) + dx (x ) 1 − η(x ) ψ (x ) . θ θ q, σ σ ,ξ ,ξ R R Recalling the hypothesis (4.2), the explicit expression (1.1)for ψ , and that we are σ,ξ assuming q > 0, from the above results we derive 2 2 ψ ψ 1 − 2 ≤ dx (x ) q, σ,ξ L (R ) σ ξ + , −∞ 2 2 + dx 1 − η(x ) ψ (x ) q, ,ξ R ∩{|x −q| > q} (x +q) 1 − 2| σ | ≤ √ dx e 2 π  | σ | (x −q) 2| σ | + 1 + ∞ dx e q, L (R ) {|x −q| > q} q ∞ − − 2 2 2| σ | 2| σ | ≤ e dx e 2 π  | σ | 0 2 2 2 q (x −q) − − 2 2 4| σ | 4| σ | + 1 + ∞ e dx e q, L (R ) 2 2 2 q q − − 2 2 2| σ | 4| σ | ≤ e + 2 1 + ∞ e . q, L (R ) 2 45 Page 30 of 43 C. Cacciapuoti et al. On the other hand, by arguments similar to those employed above we get 2 2 2 χ χ ψ ψ ( η − 1) = dx η(x ) − 1 (x ) q, 2 q, σ,ξ σ,ξ L (R ) (x −q) 1 − 2| | χ ∞ ≤ 1 + √ dx e q, L (R ) 2 π  | σ | η {|x −q| > q} 2 2 η q 4| σ | ≤ 2 1 + e . q, L (R ) √ √ Summing up, the previous results and the basic relation a + b ≤ a + b for a, b > 0imply 2 2 2 q q 1 − − 2 5/4 2 σ σ 4| | 8| | χ ∞ − ψ ≤ √ e + 2 1 + 2 q, L (R ) σ σ,ξ + ,ξ L (R ) 1 − ε 5/4 2 | σ | ≤ √ + 2 1 + e , q, L (R ) which suffices to infer the thesis on account of the uniform boundedness of η. q, Example 4.3 For ∈ (0, 1], consider the sharp cut-off functions 0if x ≤ (1 − ) q , (x ) = q, 1if x >(1 − ) q , which clearly satisfy the hypothesis of Lemma 4.2. The corresponding elements ∈ L (R ) defined according to Eq. (4.3) consist of normalized truncations σ ξ of the coherent state ψ and fulfill the condition (1.14) as a consequence. σ,ξ η χ It is worth noting that for = 1wehave ≡ 1on R , so that the associated q,η + function  is just the re-normalization of the bare truncation ψ introduced in Eq. σ,ξ σ,ξ (1.13), i.e., = ψ 2 . σ σ L (R ) σ,ξ ,ξ ,ξ + Example 4.4 For ∈ (0, 1/2), consider the smooth functions on R such that 0for |x − q| >(1 − ) q , χ χ (x ) = (x ) ≤ 1 . η η q, q, 1for |x − q| < q , Again, the assumptions of Lemma 4.2 are certainly verified and the related functions have compact support in R , besides satisfying the bound (1.14). ,ξ The semiclassical limit on a star-graph with Kirchhoff… Page 31 of 43 45 In addition to states fulfilling the requirements of Definitions 1.1 and 1.3, our arguments will often involve the unnormalized element ⎛ ⎞ σ ,ξ ⎜ ⎟ ⎜ ⎟ ∈ L (G),  ≡ , (4.4) ⎜ ⎟ σ ,ξ σ ,ξ . 0 0 ⎝ . ⎠ along with its classical counterpart ⎛ ⎞ ,x ⎜ ⎟ ⎜ ⎟ ∈ L (G),  ≡ ⎜ ⎟ , (4.5) σ σ ,x ,x . 0 0 ⎝ . ⎠ with defined as in Eq. (1.13) and σ ,ξ φ (ξ) := ψ (x). σ ,x σ ,ξ 0 0 4.1 Comparing the dynamics. Proof of Theorem 1.4 Let  and  (ξ) be, respectively, as in Eqs. (4.4) and (4.5), and note that from σ ,ξ σ ,(·) 0 0 the triangular inequality it follows t i −i H  A it L K t K e  − e e  (ξ) σ σ ,ξ ,(·) 0 t L (G) t t −i H  −i H K K ≤ e  − e σ ,ξ σ ,ξ 0 0 L (G) t i −i H  A it L K t K + e  − e e  (ξ) σ σ ,ξ ,(·) 0 t L (G) i i A it L  A it L t K t K + e e  (ξ) − e e  (ξ) . (4.6) σ ,(·) σ ,(·) t t L (G) −i H Regarding the first term on the right-hand side of Eq. (4.6), by the unitarity of e and the condition (1.14) we infer t t −i H  −i H K K e  − e  =  − 2 2 σ ,ξ σ ,ξ σ ,ξ σ ,ξ 0 0 L (G) 0 0 L (G) − ε =  − ψ ≤ C e . 2 0 σ ,ξ σ ,ξ 0 0 L (R ) As for the second term in Eq. (4.6), note that Eqs. (2.9) and (3.21)give t i −i H  A it L K t K e  − e e  (ξ) σ ,ξ σ ,(·) 0 t 45 Page 32 of 43 C. Cacciapuoti et al. ⎛   ⎞ − A it L t 0 U ψ − e e (0⊕φ ) (ξ) t σ ,ξ σ ,(·) 0 t ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + A it L U ψ − e e (0⊕φ ) (− ξ) t σ σ ,ξ ,(·) 0 t ⎜ ⎟ ⎜ ⎟ − S . ⎜ ⎟ ⎝ . ⎠ it L Since e (0⊕φ ) (± ξ) = φ (±ξ ) = ψ (x ) for x ∈ R , from the above σ ,x σ ,x t σ ,±ξ t t t identity and from Lemma 4.1 we deduce t i −i H  A it L K t K e  − e e  (ξ) σ ξ σ , ,(·) L (G) 0 t − A ≤ 2 U ψ − e ψ σ σ 2 t ,ξ ,ξ 0 t t L (R ) 2 2 2 + A 2σ + 2 |S | U ψ − e ψ ≤ 2 e . σ σ 2 t ,ξ ,−ξ 0 t t L (R ) =1 Let us finally consider the third term in Eq. (4.6). Recalling again the identity (3.21), we obtain it L  it L K K e  (ξ) − e  (ξ) σ σ ,(·) ,(·) t t ⎛     ⎞ it L it L 0  0 e (0⊕φ ) (ξ) − e (0⊕ ) (ξ) ,(·) σ ,(·) t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛     ⎞ it L it L 0  0 e (0⊕φ ) (− ξ) − e (0⊕ ) (− ξ) σ ,(·) σ ,(·) ⎜ ⎟ ⎜ ⎟ − S⎜ ⎟ . ⎝ . ⎠ From the above identity, by arguments similar to those employed previously we get it L  it L K K e  (ξ) − e  (ξ) σ ,(·) σ ,(·) t t L (G) Note also that, on account of Eq. (3.23), we have n n 2 2 n − 2 2 2 2 2 1 + |S | = 1 +|S | + |S | = 1 + + (n − 1) = 2 . 1 11 1 n n =1 =2 The semiclassical limit on a star-graph with Kirchhoff… Page 33 of 43 45 ≤ 2 ψ − σ ,ξ σ ,ξ t t L (R ) t t + − 2 ε 2  2 2 |σ | + 2 |S | ψ −  ≤ 4C e . 1 2 σ ,−ξ σ ,−ξ 0 t t L (R ) t t + =1 Summing up, the above bounds imply the thesis. 4.2 Comparing the wave and scattering operators. Proof of Theorem 1.6 T T Note the identity S (1, 0,..., 0) = (1 − 2/n, −2/n,... , −2/n) and recall the expression of  given in Eq. (2.13). Then, by simple computations we get ( = 1,..., n) 1 − (1 ∓ F F )  if  = 1 , c σ ,ξ (  ) = ,ξ − (1 ∓ F F ) if  = 1 . c σ ,ξ Additionally, recalling the expression of  given in Eq. (3.24), we get cl 1 − (∓p)  (ξ) if  = 1 , ⎨ θ σ ,x (  ) (ξ) = cl σ ,x − (∓p)  (ξ) if  = 1 . σ ,x In view of these results together with the identity (1.15) we derive ±   ∗ − (  )(ξ) = √ 1 ∓ F F − 2 (∓ p) 2 s 2 σ ,ξ cl σ ,(·) c σ ,ξ 0 0 L (G) 0 L (R ) = √ (1 − 2 θ(∓ p))F ∓ F  . c s 2 σ ,ξ 0 L (R ) By the bound (1 − 2 (∓p))F ∓ F ≤|1 − 2 (∓p)| ≤ 2 and by Eq. θ c s θ c s (1.14), we infer (1 − 2 (∓p))F ∓ F c s 2 σ ,ξ L (R ) 0 + − ε ≤ 2C e + (1 − 2 (∓p))F ∓ F ψ . 0 c s 2 σ ,ξ 0 L (R ) In what follows we prove the following upper bound which concludes the proof of the theorem 2 2 σ p √ − − 4σ (1 − 2 (∓p))F ∓ F ψ ≤ 2 e + e . (4.7) θ c s 2 σ ,ξ 0 L (R ) + 45 Page 34 of 43 C. Cacciapuoti et al. We start with the identity (1 − 2 (∓p))F ∓ F ψ θ c s σ 2 ,ξ 0 L (R ) ∞ ∞ = dk dx 1 − 2 (∓p) cos(kx ) ± i sin(kx ) ψ (x ) . σ ,ξ 0 0 Considering separately the cases p > 0 and p < 0 for the two possible choices of the signs, it is easy to convince oneself that (1 − 2 (∓p))F ∓ F ψ θ c s σ 2 ,ξ 0 L (R ) ∞ ∞ ⎪ ikx dk dx e ψ (x ) if p > 0 , σ ,ξ 0 0 ∞ ∞ ⎪ −ikx dk dx e ψ (x ) if p < 0 . σ ,ξ 0 0 Recall that the Fourier transform of ψ is given by σ ,ξ 1/4 1 √ 2 2 2 −ikx  −σ (k−p/) −ikq Fψ (k) := √ dx e ψ (x ) = σ e . σ ,ξ σ ,ξ 0 0 2 R Let us assume p > 0, we have the chain of identities/inequalities 1/2 ∞ ∞ ikx dk dx e ψ (x ) = 2 F ψ (−·) σ ,ξ σ ,ξ 0 0 L (R ) 0 0 ≤ 2 F (−·) ,ξ L (R ) + 2 F(1 − )ψ (−·) . σ ,ξ L (R ) 0 + Reasoning like for the bound in Eq. (4.1), we obtain 1/2 2 2 2 p 1 0 Fψ (−·) = dk Fψ (−k) ≤ √ e σ ,ξ σ ,ξ 0 L (R ) 0 and F(1 − )ψ (−·) ≤ F(1 − )ψ θ θ 2 2 σ ,ξ σ ,ξ 0 L (R ) 0 L (R) = (1 − )ψ σ ,ξ 0 L (R) 1/2 q 2 − 4σ = dx ψ (x ) ≤ √ e , ,ξ −∞ which conclude the proof of the bound (4.7)for p > 0. The proof of the bound for p < 0 is identical and we omit it. The semiclassical limit on a star-graph with Kirchhoff… Page 35 of 43 45 Identity (1.18) follows immediately from Eqs. (1.7) and (1.12). 5 Final remarks 5.1 A comparison with different approaches to the definition of a classical dynamics on the graph Our approach to the semiclassical limit was inspired by Hagedorn’s work [12]. In general, a coherent state (on the real-line) is the wave function ψ : R → C defined as σ ˘ 1 i ψ ψ σ σ ˘ (x ) = ( , , q, p; x ) := √ exp − (x − q) + p(x − q) , 1/4 (2 π ) σ 4 2 −1 −2 σ σ ˘ σσ ˘ σ with (p, q) ∈ R and , ∈ C\{0} such that Re( ) =| | > 0. In his seminal paper [12], Hagedorn provides the semiclassical evolution of a coher- ent state in the presence of a regular (at least C (R)) interaction potential V.By one of the main results in [12], the quantum evolution of the coherent state ψ is close (with respect to the L (R)-norm and for  small enough) to the wave function ψ σ σ ˘ x → e ( , , q , p ; x ), where the pair (q , p ) is the solution of the Hamilton t t t t t t equations ⎪ q ˙ = p , t t p ˙ =−V (q ), t t (q , p ) = (q, p), 0 0 the pair (σ , σ ˘ ) is given by t t ∂q i ∂q ∂ p ∂ p t t t t σ = σ + σ ˘ , σ ˘ =−2i σ + σ ˘ t t ∂q 2 ∂ p ∂q ∂ p and A is the classical action t 2 A = ds − V (q ) . t s 2m In our notation, one can associate with the quantum state ψ the phase space function (q, p) → (φ (σ, σ ˘ , x ))(q, p) := ψ (σ, σ ˘ , q, p; x ). By this correspondence, one gets it L ψ (σ , σ ˘ , q , p ; x ) = e φ (σ , σ ˘ , x ) (q, p), where L denotes the Liouville t t t t t t V operator associated with the vector field of the classical Hamiltonian + V (q);this 2m it L is analogous to e  (ξ), ξ ≡ (q, p). σ ,x We remark that, unlike the case of a quantum particle in the presence of a regular potential, in general there is no trajectory of a classical particle which describes the 45 Page 36 of 43 C. Cacciapuoti et al. −i H semiclassical limit of a quantum evolution of the form e  . As a conse- σ ,ξ quence, the semiclassical dynamics is not described by the Hamilton equations. One way to overcome this difficulty is to assign a probability to every possible path on the graph. Typically, the probability of a certain path is postulated, and given in terms of the square modulus of the quantum transition (or stability) amplitudes (see, e.g., [3, Sec. II.A] or [4, Sec. 6.1]). For a star-graph the latter coincide with the elements of the (vertex) scattering matrix, defined for generic boundary conditions, e.g., in [16, Thm. 2.1] or [4, Lem. 2.1.3]. For Kirchhoff boundary conditions the elements of the scattering matrix are given by − , ,  = 1,..., n,see [3, Eq. (1)] (for the star-graph C = 1), and [4, Ex. 2.1.7, p. 41]. This is the approach used (for compact bb graphs) by Kottos and Smilansky in [17] and in several other works, see, e.g., [3], the review [14], and the monograph [4]. We have already noted that, up to a sign, the coefficients − coincide with the elements of the matrix S identifying both the classical and quantum scattering operators. Compared to this approach, we followed a different train of thought, starting from the trivial dynamics of a classical particle on the half-line with elastic collision at the origin and making use of a Kre˘ın-type resolvent formula (see [20,21]) to find a suitable singular perturbation of the Liouville operator associated with such a trivial dynamics. We remark that, in a similar way, one can reconstruct the Hamiltonian H starting from the Hamiltonian of a quantum particle on the half-line with Dirichlet boundary conditions at the origin. We defined the generator of the trivial dynamics on the half-line through Eq. (1.8). −it L Note that if f ∈ dom(L ), then f = e f satisfies the Liouville equation D t i f = L f , t D ∂t but the action of the group can be extended in a natural way to any bounded function. Since the evolution is unitary in L (R × R),if 2 = 1, we can interpret L (R ×R) −it L (q, p) := e f (q, p) as a probability density in the phase space R × R. Setting (q, p) := | f (q, p)| ,for all t ∈ R one has that pt pt ⎨ q − , p if q − > 0 m m (q, p) = (5.1) pt pt − q + , −p if q − < 0 , m m and it satisfies the equation ρ ρ i =−iX ·∇ , t 0 t ∂t pt for all f ∈ C (R × R) and q − = 0. m The semiclassical limit on a star-graph with Kirchhoff… Page 37 of 43 45 We remark that, assuming elastic collision at the origin, a classical particle moving on the half-line follows a simple, though discontinuous, trajectory in the phase space: at any time t ∈ R any initial state (q, p) ∈ R × R is mapped to pt pt ⎨ q + , p if q + > 0 , m m (q, p) := (5.2) pt pt − q − , −p if q + < 0 . m m Hence, given a density : R × R → R in the phase space, one has the identity + + −it L ρ ρ ϕ (q, p) = ( (q, p)) [see Eqs. (5.1) and (5.2)]. In this sense, the group e t −t given in Eq. (1.8) describes a classical particle on the half-line. The function f := −it L e f should be interpreted as a classical wave function, with associated probability density function in the phase space given by (q, p) := f (q, p) . t t −it L On a graph this interpretation fails when the generator of the dynamics is e . In particular, from Eq. (1.9), for all t ∈ R it follows ⎪ pt pt f q − , p if q − > 0 , m m −it L e f (q, p) = ⎪ pt pt (S) f − q + , −p if q − < 0 , ⎪ , m m −it L but the density (q, p) = e f (q, p) cannot be understood in terms of a ,t trajectory of a classical particle since in this case there is no phase space flow such ρ ρ ϕ that (q, p) = ( (q, p)). Also, such a density does not coincide with the time t −t evolution of the initial density | f | as prescribed by Barra and Gaspard, see [3,Eq. (10)]. The latter, adapted to our setting and notation, and taking into account the fact that we are considering a non-compact graph, would give (for all t ∈ R) pt pt f q − , p if q − > 0 , m m BG (q, p) = ,t pt pt ⎪ |(S) | f − q + , −p if q − < 0 . m m Additionally, we remark that the initial state  (ξ) does not define a probability σ,x density by the relation (ξ) =  (ξ) , because, in general, it does not belong σ,x to L (R × R). Nevertheless, the phase space evolution of the approximated coherent states  (ξ) σ,x induced by the Liouville operator L turns out to be a useful tool for the investigation of the semiclassical limit of the quantum evolution on the graph. 45 Page 38 of 43 C. Cacciapuoti et al. 5.2 Coherent states on a star-graph with an even number of edges Obviously, by superposition, one could consider initial states of the form ⎛ ⎞ σ ,ξ ⎜ ⎟ ⎜ ⎟ ≡ , σ σ ( ,ξ ),...,( ,ξ ) . 1 1 n n ⎝ ⎠ σ ,ξ n n with σ > 0 and ξ = (q , p ) ∈ R × R, and results similar to the ones stated in Theorems 1.4 and 1.6 hold true (with additive error terms). For n even it is possible to construct states that propagate exactly like coherent states on the real line. Suppose n even and consider a state ψ defined component-wisely by σ,ξ ψ if  = 1,..., n/2 , σ,ξ (ψ ) := σ ξ ψ if  = n/2 + 1,..., n . ,− ξ It is easy to check that such states belong to the domain of H . Next, consider the state e ψ . Taking the time derivative component by component one has σ ,ξ ∂ i   i A A t t ˚ ˚ i  e (ψ ) =− e (ψ ) , σ ,ξ σ ,ξ t t t t ∂t 2m t ˚ by the definition of coherent states, see Eq. (1.1). Since e ψ ∈ dom(H ) the σ ,ξ latter is equivalent to ∂ i i A A t t ˚ ˚ i  e ψ = H e ψ . σ ,ξ σ ,ξ t t t t ∂t i   t  i A −i H A t ˚ ˚ K ˚ t ˚ Moreover e ψ = ψ . Hence, e ψ = e ψ . On the other σ ,ξ σ ,ξ σ ,ξ σ ,ξ t t t =0 0 0 t t ˚ ˚ ˚ hand, define the classical state φ , component-wisely, by (φ ) (ξ) := (ψ ) (x ). σ,x σ,x σ,ξ ˚ ˚ Noting the identity − (φ ) (− ξ) = (φ ) (ξ), and by using Eq. δ, σ,x σ,x =1 n it L K ˚ ˚ ˚ (1.9), it is easy to infer the identity e φ (ξ) = φ (ξ ) = ψ , so that σ ,(·) σ ,(·) t σ ,ξ t t t t t i −i H A it L K ˚ K ˚ e ψ = e e φ (ξ), σ ,ξ σ ,(·) 0 t which is equivalent to (1.2). In this sense, up to a normalization factor 2/n, states of the form ψ are coherent states on a star-graph with an even number of edges. σ,ξ Funding Open Access funding provided by Università degli Studi dell’Insubria Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. The semiclassical limit on a star-graph with Kirchhoff… Page 39 of 43 45 Compliance with ethical standards Conflict of interest No potential competing interest was reported by the authors. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. A Appendix: Wave operators for the pair (Dirichlet Laplacian, Neumann Laplacian) on the half-line Proposition A.1 Let  be the wave operators for the pair (Dirichlet Laplacian, ND Neumann Laplacian) in L (R ), defined by ± N D := s-lim U U . t →±∞ −t t ND There holds true: ± ∗ =±F F . ND c Proof We use a density argument. Let ψ ∈ L (R ). For any ε > 0 there exists ∈ D/N C (R ) such that ψ − ≤ ε /4. Recalling the trivial bounds ≤ 1 L (R ) t 0 + and ≤ 1, we infer s/c N D ∗ N D ∗ (U U ∓ F F ) ψ 2 ≤ + (U U ∓ F F ) 2 . s s L (R ) L (R ) −t t c −t t c + + Hence, it is enough to prove that N D ∗ ∞ ϕ ϕ lim (U U ∓ F F ) = 0 ∀ ∈ C (R ). (A.1) s + −t t c L (R ) 0 t →±∞ Note that N D ∗ D N ∗ ϕ ϕ (U U ∓ F F ) (U ∓ U F F ) 2 2 s s −t t c L (R ) t t c L (R ) + + D ∗ N ∗ (U F ∓ U F )F 2 . t s t c L (R ) Moreover, for all t ∈ R the following identities hold true: 2i 2 D −ik t (U ψ)(x ) = √ dk e sin(kx)(F ψ)(k) ; 2 π 0 45 Page 40 of 43 C. Cacciapuoti et al. N −ik t (U ψ)(x ) = dk e cos(kx)(F ψ)(k). 2 π 0 Hence, 2i 2 D ∗ −ik t ψ ψ (U F )(x ) = √ dk e sin(kx ) (k), t s 2 π 2 2 N ∗ −ik t (U F ψ)(x ) = √ dk e cos(kx ) ψ(k), t c 2 0 which give 2 2 D ∗ N ∗ −ik t ϕ ϕ (U F ∓ U F )F (x ) = √ dk (i sin(kx )∓cos(kx )) e (F )(k) s s t s t c 2 0 2 2 ∓ikx −ik t =∓ √ dk e (F )(k). 2 π We have obtained the following explicit formula for the quantity we are interested in ∞ ∞ 2 2 N D ∗ 2 ∓ikx −ik t ϕ ϕ (U U ∓ F F ) = dx dk e (F )(k) . s s −t t c L (R ) 0 0 We note that, to prove the statement for  we have to study the limit t →+∞ ND of ∞ ∞ 2 2 N D ∗ 2 −ikx −ik t ϕ ϕ (U U − F F ) = dx dk e (F )(k) , (A.2) s s −t t c L (R ) 0 0 while, to prove the statement for  we have to study the limit t →−∞ of ND ∞ ∞ N D ∗ 2 ikx +ik |t | ϕ ϕ (U U + F F ) = dx dk e (F )(k) . s 2 s −t t c L (R ) 0 0 In what follows we focus the attention on the limit t →+∞. The other limit is obtained with trivial modifications. Hence, from now on we assume t > 0. From Eq. 1/2 1/2 (A.2), changing variables k → = kt and x → y = x /t , we obtain ∞ ∞ 2 2 N D ∗ 2 −i η y−i η 1/2 ϕ η ϕ η (U U − F F ) = dy d e (F )( /t ) s s −t t c 2 L (R ) 1/2 π t 0 0 1 ∞ 2 2 1 2 2 = dy |F (y, t )| + dy |F (y, t )| , 1/2 1/2 π t π t 0 1 with η η −i y−i 1/2 η ϕ η F (y, t ) := d e (F )( /t ). 0 The semiclassical limit on a star-graph with Kirchhoff… Page 41 of 43 45 ∞ ∞ ϕ ϕ For any ∈ C (R ), F belongs to C (R ), it decays at infinity faster than + s + any polynomial in 1/k, (F )(0) = 0, moreover 2k ϕ ϕ |(F )(k)|≤ √ dx x | (x )| . 2 π Hence, dk |F (k)| < ∞ for all < 2 . (A.3) ∞ ∞ ϕ ϕ Additionally, for any ∈ C (R ), F belongs to C (R ), it decays at infinity + c + ϕ √ ϕ faster than any polynomial in 1/k, and |(F )(k)|≤ dx | (x )|. Hence, 2 π dk |F (k)| < ∞ for all < 1 . (A.4) c δ 2 2 η η i d η η −i y−i −i y−i Starting from the identity e = e ,byintegrationbyparts, η η y+2 d we obtain F (y, t ) = F (y, t ) + F (y, t), 1 2 with 2 2i η η −i y−i 1/2 η η F (y, t ) := d e (F )( /t ) 1 s (y + 2 ) and 1 2 1 η η −i y−i 1/2 η ϕ η F (y, t ) := d e F ((·) ) ( /t ). 2 c 1/2 t i (y + 2 ) 1 1 1 Since y and are both positive, from the trivial inequality ≤ for a c η η (y+2 ) y (2 ) all y, > 0 and for all a, b, c > 0 such that a = b + c, we deduce that C 1 1/2 η ϕ η |F (y, t )|≤ d |(F )( /t )| 1 s 1/4 7/4 C 1 C ξ ξ = d |(F )( )|= ; 3/8 1/4 7/4 3/8 1/4 t y ξ t y here and in the following C denotes a generic positive constant whose value may ϕ ϕ depend only on integrals of the sine (or cosine) Fourier transform of (or (·) ), as in Eqs. (A.3) and (A.4); the value of C may change from line to line and precise values for the constants can be obtained. In a similar way we infer, C 1 1/2 η ϕ η |F (y, t )|≤ d F ((·) ) ( /t ) 2 c 1/2 1/4 3/4 t y 0 45 Page 42 of 43 C. Cacciapuoti et al. C 1 C = d ξ F ((·) ) (ξ) = . 3/8 1/4 3/4 3/8 1/4 t y ξ t y Hence, 1 1 2 C 1 C | | dy F (y, t ) ≤ dy ≤ . 1/2 5/4 1/2 5/4 t t y t 0 0 On the other hand, for y > 1, ∞ ∞ C 1 C 1 C 1/2 η ϕ η ϕ |F (y, t )|≤ d |(F )( /t )|= d ξ |(F )(ξ)|= 1 s s y y ξ y 0 0 and ∞ ∞ C C C 1/2 η ϕ η ϕ |F (y, t )|≤ d F ((·) ) ( /t ) = d ξ F ((·) ) (ξ) = . 2 c c 1/2 t y y y 0 0 So, we obtain ∞ ∞ 2 C 1 C dy |F (y, t )| ≤ dy ≤ . 1/2 1/2 2 1/2 t t y t 1 1 In this way we have proved that, for all ∈ C (R ) there exists a constant C such that N D ∗ (U U − F F ) 2 ≤ for all t > 1 , s L (R ) −t t c + 1/4 and the latter claim gives the limit in Eq. (A.1)for t →+∞. References 1. Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Fast solitons on star graphs. Rev. Math. Phys. 23(04), 409–451 (2011) 2. F. Barra, P. Gaspard: Transport and dynamics on open quantum graphs. Phys. Rev. E 65 (2001), 016205 (21 pages) 3. F. Barra, P. Gaspard: Classical dynamics on graphs. Phys. Rev. E 63 (2001), 066215 (22 pages) 4. Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Mono- graphs, vol. 186. American Mathematical Society, Providence, RI (2013) 5. Cacciapuoti, C., Fermi, D., Posilicano, A.: The semi-classical limit with a delta potential. Annali di Matematica Pura ed Applicata, (2020). (37 pages) https://doi.org/10.1007/s10231-020-01002-4 6. Chernyshev, V.L.: Time-dependent Schrödinger equation: statistics of the distribution of Gaussian packets on a metric graph. Tr. Mat. Inst. Steklova 270, 249–265 (2010) (Russian). Translation in Proc. Steklov Inst. Math. 270(1), 246–262 (2010) 7. Chernyshev, V.L., Shafarevich, A.I.: The semiclassical spectrum of the Schrödinger operator on a geometric graph. Mat. Zametki 82(4), 606–620 (2007) (Russian). Translation in Math. Notes 82(3–4), 542–554 (2007) The semiclassical limit on a star-graph with Kirchhoff… Page 43 of 43 45 8. Chernyshev, V.L., Shafarevich, A.I.: Semiclassical asymptotics and statistical properties of Gaussian packets for the nonstationary Schrödinger equation on a geometric graph. Russ. J. Math. Phys. 15(1), 25–34 (2008) 9. Chernyshev, V.L., Shafarevich, A.I.: Statistics of Gaussian packets on metric and decorated graphs. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2007), 20130145 (11 pages) (2014) 10. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin (2000) 11. Gaveau, B., Okada, M., Okada, T.: Explicit heat kernels on graphs and spectral analysis. In: Fornaess J.E. (ed.) Several Complex Variables. Proceedings of the Mittag-Leffler Institute, Stockholm, 1987– 1988, Princeton Mathematical Notes, vol. 38. Princeton University Press (1993) 12. Hagedorn, G.A.: Semiclassical quantum mechanics I. The  → 0 limit for coherent states. Comm. Math. Phys. 71, 77–93 (1980) 13. Halmos, P.R., Sunder, V.S.: Bounded Integral Operators on L Spaces. Springer-Verlag, Berlin-New York (1978) 14. Gnutzmann, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006) 15. Kostrykin, V., Schrader, R.: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In: Berkolaiko, G., Carlson, R., Fulling, S., Kuchment, P. (eds.) Quantum Graphs and Their Applications. Contemporary Mathematics, vol. 415, pp. 201–226. American Math. Society, Providence, RI (2006) 16. Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32(4), 595–630 (1999) 17. Kottos, T., Smilansky, U.: Quantum chaos on graphs. Phys. Rev. Lett. 79(24), 4794–4797 (1997) 18. Kottos, T., Smilansky, U.: Chaotic scattering on graphs. Phys. Rev. Lett. 85(5), 968–971 (2000) 19. Maslov, V.P.: The Complex WKB Method for Nonlinear Equations I. Linear Theory. Birkhäuser Verlag, Basel (1994) 20. Posilicano, A.: A Kre˘ın-like formula for singular perturbations of self-adjoint operators and applica- tions. J. Funct. Anal. 183, 109–147 (2001) 21. Posilicano, A.: Self-adjoint extensions of restrictions. Oper. Matrices 2(4), 483–506 (2008) 22. Weder, R.: Scattering theory for the matrix Schrödinger operator on the half line with general boundary conditions. J. Math. Phys. 56(9), 092103 (24 pages) (2015). Erratum J. Math. Phys. 60(1), 019901 (1 page) (2019) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

The semiclassical limit on a star-graph with Kirchhoff conditions

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Abstract

We consider the dynamics of a quantum particle of mass m on a n-edges star-graph −1 2 with Hamiltonian H =−(2m)   and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre˘ın’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators ⊕ ⊕ for the couple (H , H ), where H is the Hamiltonian with Dirichlet conditions in D D the vertex. Keywords Semiclassical dynamics · Quantum graphs · Coherent states · Scattering theory Mathematics Subject Classification 81Q20 · 81Q35 · 47A40 The authors acknowledge the support of the National Group of Mathematical Physics (GNFM-INdAM). B Claudio Cacciapuoti claudio.cacciapuoti@uninsubria.it Davide Fermi fermidavide@gmail.com https://fermidavide.com Andrea Posilicano andrea.posilicano@uninsubria.it DiSAT, Sezione di Matematica, Università dell’Insubria, via Valleggio 11, 22100 Como, Italy Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Present address: Dipartimento di Matematica ‘Guido Castelnuovo’, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Roma, Italy 0123456789().: V,-vol 45 Page 2 of 43 C. Cacciapuoti et al. 1 Introduction Aim of this work is to provide the semiclassical dynamics and scattering for an approx- imate coherent state propagating freely on a star-graph, in the presence of Kirchhoff conditions in the vertex. Since the pioneering work of Kottos and Smilansky [17], having in mind applica- tions to quantum chaos, the semiclassical limit of quantum graphs is often understood as the study of the distribution of eigenvalues (or resonances, see [18]) of self-adjoint −1 2 realizations of −(2m)   on the graph. To the best of our knowledge, a first study of the semiclassical limit for quantum dynamics on graphs is due to Barra and Gaspard [2](seealso[3], where the limiting classical model is comprehensively discussed). In this case, the semiclassical limit is understood in terms of the convergence of a Wigner-like function for graphs when (the reduced Planck constant) goes to zero. Inspired by the work of Hagedorn [12], instead, we look directly at the dynamics of the wave-function, for a class of initial states which are close to Gaussian coherent states supported on one of the edges of the graph. Closely related to our work is a series of papers by Chernyshev and Shafarevich [6, 8,9] in which the authors study the  → 0 limit of Gaussian wave packets propagating on graphs. Their main interest is the asymptotic growth (for large times) of the number of wave packets propagating on the graph. The main tool for the analysis is the complex WKB method by Maslov (see [19]). We also point out the work [7], by the same authors, in which they study the small  asymptotics of the eigenvalues of Schrödinger operators on quantum graphs (with Kirchhoff conditions in the vertices and in the presence of potential terms). In our previous work [5] we studied the semiclassical limit in the presence of a singular potential. Specifically, we considered the operator H , which is the quantum Hamiltonian in L (R) formally written as H =−  + α , where m is the mass α δ0 2m of the particle, is the Dirac-delta distribution centered in x = 0, and α is a real δ0 constant measuring the strength of the potential. Given a Gaussian coherent state on the real line of the form 1 1 i ψ (x ) := √ exp − (x − q) + p(x − q) for x ∈ R , σ,ξ 1/4 π σ 4σ σ (2 ) (1.1) with σ ∈ C,Re σ = σ > 0 and ξ ≡ (q, p) ∈ R , we studied the limit  → 0of −i H e . ,ξ To this aim we reasoned as follows. For fixed x ∈ R, consider the classical wave function defined by φ : R → C , φ (ξ) := ψ (x). σ,x σ,x σ,ξ Consider the vector field X (q, p) = (p/m, 0) associated with the free classical Hamiltonian h (q, p) = p /(2m) (q is the position and p the momentum of the 0 The semiclassical limit on a star-graph with Kirchhoff… Page 3 of 43 45 classical particle of mass m), and the Liouville operator p ∂ ∞ 2 dom(L) := C (R ), L := −iX ·∇ = − i . m ∂q Set pt p t it ξ := q + , p , A := , σ := σ + , t t 0 m 2m 2mσ where ξ is the solution of the free Hamilton equations, A is the (free) classical action, t t and σ takes into account the spreading of the wave function. If the dynamics is free (i.e., α = 0), one has the identity t i −i H  A 0 t e ψ (x ) = e ψ (x). σ ,ξ σ ,ξ 0 t The latter can be rewritten as t i −i H  A it L 0 t 0 e ψ (x ) = e e φ (ξ), (1.2) σ ,ξ ,x 0 t it L ∞ 2 2 2 where e is the realization in L (R ) of the strongly continuous (in L (R )) group of evolution generated by the self-adjoint operator L = L; explicitly, one it L has e f (ξ) = f (ξ ). ◦ ∞ Since H is a self-adjoint extension of H := H  C (R\{0}), mimicking the α 0 t i −i H  A it L α t β identity (1.2), we compared e ψ (x ) with e e φ (ξ), with L a σ ,x σ ,ξ 0 t ∞ 2 self-adjoint extension of L := L  C (M ), M := R \{(0, p) | p ∈ R}. Here, β 0 0 0 0 c is a real constant which parameterizes the self-adjoint extension, and it turns out that the optimal choice is β = 2 α / (see [5] for the details). In the same spirit, in the present work, we study the small  asymptotic of −i H e  where H is the quantum Hamiltonian defined as the self-adjoint real- σ ,ξ −1 2 ization of −(2m)   on the star-graph with Kirchhoff conditions in the vertex, and resembles a coherent state concentrated on one edge of the graph (see Sect. 1.3 σ,ξ below for the precise definition). The paper is structured as follows. In the remaining sections of the introduction we give the main definitions and results. Sections 2 and 3 contain a detailed description of the quantum and semiclassical dynamics on the star-graph respectively. In Sect. 4 we give the proofs of Theorems 1.4 and 1.6. Section 5 contains some additional remarks and comments. The paper is concluded by an “Appendix A” in which we present a proof of a technical result, namely an explicit formula for the wave operators for the pair (Dirichlet Laplacian, Neumann Laplacian) on the half-line. 1.1 Quantum dynamics on the star-graph By star-graph we mean a non-compact graph, with n edges (or leads) and one vertex. Each edge can be identified with a half-line, the origins of the half-lines coincide and identify the only vertex of the graph. 45 Page 4 of 43 C. Cacciapuoti et al. We recall that the Hilbert space associated with the star-graph is L (G) ≡ 2 2 ⊕ L (R ), with the natural scalar product and norm; in particular, for the L -norm =1 we use the notation 1/2 1/2 2 2 2 ≡ dx |ψ(x )| := dx | (x )| . L (G) G 0 =1 2 2 If ψ ∈ L (G), ψ ∈ L (R ) is its -th component with respect to the decomposi- n 2 tion ⊕ L (R ). In a similar way one can define the associated Sobolev spaces; in =1 2 n 2 particular, we set H (G) ≡⊕ H (R ), with the natural scalar product and norm. =1 We are primarily interested in the semiclassical limit of the quantum dynamics generated by the Kirchhoff Laplacian on the star-graph, which is the operator dom(H ) := ψ ∈ H (G) ψ (0) = ··· = ψ (0), ψ (0) = 0 , (1.3) K 1 n =1 H ψ := − ψ ; (1.4) 2m here ψ denotes the element of L (G) with components ψ , and ψ(0) (resp., ψ (0))the ψ ψ vector in C with components (0) (resp., (0)). Functions in dom(H ) are said to satisfy Kirchhoff (also called Neumann, or standard, or natural) boundary conditions. In the analysis of the semiclassical limit of the wave and scattering operators, we will have to fix a reference dynamics on the star-graph. To this aim we will consider the operator H [see also the equivalent definition in Eqs. (2.5)–(2.6) below] ⊕ 2 dom(H ) := ψ ∈ H (G) ψ (0) = ··· = ψ (0) = 0 , 1 n H ψ := − ψ ; 2m we remark that H can be understood as the direct sum of n copies of the Dirichlet Hamiltonian on the half-line (see Sect. 2.2 below for further details). We recall that the quantum wave operators and the corresponding scattering operator on L (G), are defined by t t ⊕ ± i H −i H := s-lim e e , (1.5) t →±∞ + ∗ − S := ( )  . (1.6) These operators can be computed explicitly (see Proposition 2.2 and Remark 2.3 below), and component-wisely for  = 1,..., n they read as follows: The semiclassical limit on a star-graph with Kirchhoff… Page 5 of 43 45 ± ∗ ψ = − (1 ∓ F F ) ψ , , s =1 where is the Kronecker delta, F and F are the Fourier-sine and Fourier-cosine , s c transforms respectively [see Eqs. (2.11) and (2.12)]; (S ψ) = − ψ . (1.7) =1 1.2 Semiclassical dynamics on a star-graph The generator of the semiclassical dynamics on the star-graph is obtained as a ∂ n 2 self-adjoint realization of the differential operator − i in ⊕ L (R × R), =1 m ∂q (q, p) ∈ R × R. To recover it we will make use of the method to classify the singular perturbations of self-adjoint operators developed by one of us in [20](seealso[21]). To do so, the first step is to identify a simple dynamics on the star-graph, more precisely its generator. We shall consider classical particles moving on the edges of the graph with elastic collision at the vertex. We start by considering the dynamics of a classical particle on the half-line with elastic collision at the origin. We obtain its generator as a limiting case from our previous work [5] and denote it by L . We postpone the precise definition of L to D D Sect. 3.1. Here we just note few facts. 2 2 L : dom(L ) ⊂ L (R × R) → L (R × R) is self-adjoint and acts on elements D D + + of its domain as p ∂ f (L f )(q, p) =− i (q, p) for (q, p) ∈ R × R . D + m ∂q For all t ∈ R, the action of the unitary evolution group associated with it is explicitly given by pt pt ⎨ f q + , p if q + > 0 , it L m m e f (q, p) = (1.8) pt pt − f − q − , −p if q + < 0 . m m The (trivial) classical dynamics of a particle on the star-graph with elastic collision at the vertex can be defined in the following way. Denote by f a function of the form ⎛ ⎞ f (q, p) ⎜ ⎟ n 2 . f ∈⊕ L (R × R), f (q, p) ≡ . . ⎝ ⎠ =1 f (q, p) n 45 Page 6 of 43 C. Cacciapuoti et al. If n = 1, | f (q, p)| dqdp can be interpreted as the probability of ⊕ L (R ×R) =1 finding a particle on the -th edge of the graph, with position in the interval [q, q +dq] and momentum in the interval [ p, p + dp]. ⊕ n Define the operator L := ⊕ L ; the associated dynamics is generated by the D =1 iL t n iL t unitary group e =⊕ e , and it is trivial in the sense that it can be fully =1 understood in terms of the dynamics on the half-line described above. We consider the map ( f ) (p) := lim f (q, p) for  = 1,..., n , q→0 defined on sufficiently smooth functions (we refer to Sect. 3 for the details). This map ⊕ ⊕ ⊕ can be extended to a continuous one on dom(L ). The operator L  ker( ) is D D symmetric; in Theorem 3.3, by using the approach developed in [20,21] we identify a family of self-adjoint extensions. Among those we select the one that turns out to be useful to study the semiclassical limit of exp(−iH t /) and denote it by L . K K We postpone the precise definition of L to Sect. 3.3, see, in particular, Remark 3.4. Here we just give component-wisely the formula for the associated unitary group, for  = 1,..., n and for all t ∈ R: pt pt ⎪ f q + , p if q + > 0 , m m it L e f (q, p) = 2 pt pt − f − q − , −p if q + < 0 . ⎪ , n m m =1 (1.9) We define the classical wave operators and the corresponding scattering operator n 2 on ⊕ L (R × R) by =1 ± it L −it L := lim e e (1.10) cl t →±∞ and + ∗ − S := ( )  . (1.11) cl cl cl These operators can be computed explicitly (see Proposition 3.8 below), and component-wisely they read as follows for  = 1,..., n ( is the Heaviside step function): f = − (∓p) f ; δ θ cl =1 (S f ) = − f . (1.12) cl  , =1 1.3 Truncated coherent states on the star-graph In general, there is no natural definition of a coherent state on a star-graph, neither there is a unique way to extend coherent states through the vertex. Since we are interested in The semiclassical limit on a star-graph with Kirchhoff… Page 7 of 43 45 initial states concentrated on one edge of the graph, we introduce the following class of initial states. We denote by ψ the unnormalized restriction of ψ [see Eq. (1.1)] σ,ξ σ ,ξ to R , namely, ψ ∈ L (R ), ψ (x ) = ψ (x ) for x > 0 . (1.13) σ,ξ σ,ξ σ,ξ On the graph we consider the quantum states defined as Definition 1.1 (Quantum states)Let σ ∈ C, with Re σ = σ > 0, and ξ = (q, p) ∈ R × R; consider any normalized function  ∈ L (R ), such that + + σ,ξ − ε | σ | − ψ ≤ C e for some C , ε > 0 . (1.14) 2 0 0 σ,ξ σ,ξ L (R ) We are primarily interested in quantum states on the star-graph of the form ⎛ ⎞ ,ξ ⎜ ⎟ ⎜ ⎟ ∈ L (G),  ≡ ⎜ ⎟ . σ,ξ σ,ξ ⎝ ⎠ Remark 1.2 In Definition 1.1 we assume that the constants C and ε do not depend on , ξ or σ. In what follows, whenever we refer to a state of the form  the constants σ,ξ C and ε are the ones given in Definition 1.1. One could choose the quantum states  (and the corresponding classical σ,ξ states given below) in a different way. As a matter of fact, for any choice of 2 2 2 2 ε σ ε σ the terms C exp(− q /( )) and C exp(− (q + pt /m) /(| | )) in the 0 0 t σ,ξ 0 bounds in Theorems 1.4 and 1.6 would be replaced by  − and σ ,ξ σ ,ξ 0 L (R ) − ψ respectively. σ ,ξ σ ,ξ t t L (R ) t t + Correspondingly, we will consider the family of classical states Definition 1.3 (Classical states) For any σ, ξ, and  as in Definition 1.1, consider σ,ξ the function  : R × R → C defined by σ,x (ξ) :=  (x). (1.15) σ,x σ,ξ We will make use of the family of classical states on the star-graph given by  ≡ ,x , 0,..., 0 . In general the functions  do not belong to L (R × R) but σ,x σ,x we will always assume that they are in L (R × R) (see Lemma 4.2, and Examples −it L 4.3 and 4.4 below). We remark that the classical operators e ,  , and S can be cl cl naturally extended to L (R × R). + 45 Page 8 of 43 C. Cacciapuoti et al. 1.4 Main results Our first result concerns the semiclassical limit of the dynamics. Theorem 1.4 Let σ > 0, ξ = (q, p) ∈ R × R and consider any initial state of the 0 + form  , together with its classical analogue  . Then, for all t ∈ R there holds σ ,ξ ,x 0 0 1/2 t i −i H  A it L K t K dx e  (x ) − e e  (ξ) σ σ ,x ,ξ 0 t 2 2 q 2 q (q+pt /m) − ε √ − − ε 2 2 σ 2 4σ | | 0 t 0 ≤ C e + 2C e + 2 e . (1.16) 0 0 Remark 1.5 Let t (ξ) := −mq/p be the classical collision time. Whenever |t − coll η η t |≤ m /| p| for some positive constant the second term on the right-hand coll 0 − ε η side of Eq. (1.16) is larger than 2C e . In the second part of our analysis we study the semiclassical limit of the wave operators and of the scattering operator. Theorem 1.6 Let σ > 0, ξ = (q, p) ∈ R × (R\{0}) and consider any state of the 0 + form  , together with its classical analogue  . Then, there hold σ ,x σ ,ξ 0 0 1/2 dx   (x ) − (  )(ξ) σ σ ,x ,ξ cl 0 0 2 2 q q 2 2 σ p √ − ε − 2 0 2 2 σ 4σ − 0 0 ≤ 2 C e + e + e , (1.17) and S  (x ) = (S  )(ξ). (1.18) cl σ σ ,x ,ξ 0 0 Identity (1.18) is an immediate consequence of Eqs. (1.7) and (1.12), and of the definitions of  and  . σ ,ξ σ ,x Remark 1.7 Equation (1.17) makes evident that   and (  )(ξ) are σ ξ cl σ , ,(·) 0 0 exponentially close (with respect to the natural topology of L (G)) in the semiclassical 2 2 2 2 + limit σ /q , /σ p → 0 for any ξ = (q, p) with q > 0 and p = 0. 0 0 As a matter of fact, it can be proved that the relation (1.17) remains valid also for p = 0 if one puts (  )(q, 0) =  (q, 0); the latter position appears to be σ ,x σ ,x cl 0 0 reasonable and is indeed compatible with the computations reported in the proof of 2 2 Proposition 3.8. Nonetheless, since exp(− σ p /) = 1 in this case, the resulting upper bound is of limited interest for what concerns the semi-classical limit. To say 2 2 more, for p = 0 and σ /q (or C ) small enough, by a variation of the arguments described in the proof of Theorem 1.6 one can derive the lower bound 1/2 ±  ± dx   (x ) − (  )(q, 0) σ ,(q,0) σ ,x cl 0 G The semiclassical limit on a star-graph with Kirchhoff… Page 9 of 43 45 2 2 q q − √ − ε 2 2 4σ σ 0 0 ≥ 1 − e − 2 C e . This shows that, as might be expected, the classical scattering theory does not provide a good approximation for the quantum analogue when p = 0. On the contrary, notice that Eq. (1.16) ensures a significant control of the error for the dynamics at any finite time t ∈ R even for p = 0. 2 The quantum theory 2.1 Dirichlet dynamics on the half-line Let us first consider the free quantum Hamiltonian for a quantum particle of mass m on the whole real line, defined as usual by 2 2 2 H : H (R) ⊂ L (R) → L (R), H ψ := − ψ , 0 0 2m 0 −i H together with the associated free unitary group U := e (t ∈ R). Correspond- ingly, let us recall that for any ∈ L (R) we have (x −y) m m 0 i 2 t (U ψ)(x ) = dy e ψ(y). (2.1) 2 π i t Let us further introduce the Dirichlet Hamiltonian on the half-line R , defined as usual by 1 2 dom(H ) := H (R ) ∩ H (R ), H ψ := − ψ , D + + D 2m D −i H and refer to the associated unitary group U := e (t ∈ R). As well known, the latter operator can be expressed as D − + U = U − U , (2.2) t t t where, in view of the identity (2.1), we introduced the bounded operators on L (R ) defined as follows for ψ ∈ L (R ) and x ∈ R : + + m m (x ±y) ± i 2 t (U ψ)(x ) := dy e ψ(y). (2.3) 2 i t Remark 2.1 Let us consider the bounded operator ψ(x ) if x > 0 , 2 2 : L (R ) → L (R), ( ψ)(x ) = 0if x < 0 , 45 Page 10 of 43 C. Cacciapuoti et al. together with its adjoint ∗ 2 2 ∗ : L (R) → L (R ), ( ψ)(x ) = ψ(x)(x ∈ R ). + + Namely,  gives the extension by zero to the whole real line R of any function on R , while  is the restriction to R of any function on R. Note that  is an isometry. In ∗ 2 ∗ fact,   is the identity on L (R ) and  is an orthogonal projector (but not the identity) on L (R); more precisely, we have ( is the Heaviside step function) ∗ 2 = 1 on L (R ), ∗ 2 2 ∗ : L (R) → L (R), (  ψ)(x ) = (x ) ψ(x)(x ∈ R). To proceed let us consider the parity operator 2 2 P : L (R) → L (R), (P ψ)(x ) = ψ(−x)(x ∈ R). Of course P is a unitary, self-adjoint involution which commutes with the free Hamil- tonian H , i.e., H P = PH . 0 0 Furthermore it can be checked by direct inspection that ran(P ) = ker( ) Using the bounded linear maps introduced above, one can express the operators defined in Eq. (2.3) as follows: − ∗ 0 + ∗ 0 ∗ 0 U =  U , U =  PU  =  U P . (2.4) t t t t t 0 ∗ 0 Recalling that (U ) = U , the above relation allow us to infer t −t − ∗ − + ∗ + (U ) = U ,(U ) = U . t −t t −t Let us finally point out that, on account of the obvious operator norms 1, = 1 and = 1, from Eq. (2.4) it readily follows ≤ 1 . 2.2 Dirichlet and Kirchhoff dynamics on the star-graph Let us now introduce the quantum Hamiltonian on the graph G, corresponding to Dirichlet boundary conditions at the vertex. This coincides with the direct sum of n copies of the Dirichlet Hamiltonian H on the half-line R , namely: D + ⊕ n n 1 2 dom(H ) := ⊕ dom(H ) = ⊕ H (R ) ∩ H (G), (2.5) D + D =1 =1 0 The semiclassical limit on a star-graph with Kirchhoff… Page 11 of 43 45 ⊕ n ⊕ 2 2 H := ⊕ H : dom(H ) ⊂ L (G) → L (G), D =1 D ⎛ ⎞ ⎜ . ⎟ H ψ := − . (2.6) ⎝ ⎠ D . 2m In view of the identity (2.2), it can be readily inferred that the corresponding unitary t ⊕ −i H group e (t ∈ R) can be expressed as t ⊕ −i H n − n + e =⊕ U −⊕ U , (2.7) =1 t =1 t where U is defined as in Eq. (2.3). To proceed let us consider the Kirchhoff Hamiltonian on the graph G. This is defined as in Eqs. (1.3)–(1.4). In what follows we denote by S the n×n matrix with components (S) := − , for ,  = 1,..., n . (2.8) , , By a slight abuse of notation we use the same symbol to denote the operator in L (G) defined by (S ψ) := (S) ψ ψ ∈ L (G). =1 By arguments similar to those given in the proof of [1, Thm. 2.1] (cf. also [11] and [15, Eq. (7.1)]) we get −i H n − n + e =⊕ U − S ⊕ U . (2.9) =1 t =1 t 2.3 The quantum wave operators and scattering operator Let us consider the wave operators and the corresponding scattering operator on L (G) respectively defined in Eqs. (1.5) and (1.6). Since H has purely absolutely continuous spectrum σ(H ) =[0, ∞),wehave K K ± 2 that  are unitary on the whole Hilbert space L (G), i.e., ± ∗ ± ( )  = 1 , which in turn ensures = 1 . (2.10) 1 ± Of course, the same identity (2.10) can be derived straightforwardly from the fact that  are defined as strong limits of unitary operators. 45 Page 12 of 43 C. Cacciapuoti et al. 2 2 2 Let us define the unitary operators F : L (R ) → L (R ) and F : L (R ) → s + + c + L (R ): 2i (F ψ)(k) := − √ dx sin(kx ) ψ(x)(k ∈ R ) ; (2.11) s + 2 π 0 (F ψ)(k) := √ dx cos(kx ) ψ(x)(k ∈ R ). (2.12) c + 2 π The wave operators can be computed explicitly. To this aim one could use the results from Weder [22] (see also references therein), with some modifications, since in [22] the reference dynamics is given by the Hamiltonian with Neumann boundary conditions. For the sake of completeness, we prefer to give an explicit derivation of the result, obtained by taking the limit t →±∞ on the unitary groups. We remark that in [22] the formulae are obtained by using the Jost functions. We have the following explicit formulae for the wave operators: Proposition 2.2 The quantum wave operators can be expressed as 1 1 ± n ∗ n ∗ = ⊕ (1 ± F F ) + S ⊕ (1 ∓ F F ). (2.13) s s =1 c =1 c 2 2 Proof By Eqs. (2.7) and (2.9) we easily obtain the identity ± n − − − + n + + + − = s-lim ⊕ (U U − U U ) + S ⊕ (U U − U U ) . (2.14) −t −t −t −t =1 t t =1 t t t →±∞ Let us find more convenient expressions for the operators on the right-hand side. Let ψ ∈ L (R ) and define ψ(x ) if x > 0 ψ(x ) if x > 0 ψ (x ) := ; ψ (x ) := . e o ψ(−x ) if x < 0 − ψ(−x ) if x < 0 ψ +ψ e o ∗ ∗ In view of Remark (2.1), we have:  ψ = ,  (ψ + ψ ) = 2 ψ, and  (ψ − e o e ψ ) = 0. Hence, see Eq. (2.4), − − ∗ 0 ∗ 0 U U ψ =  U  U  ψ −t t −t t ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) + U ( U ψ ) + U ( U ψ ) e e o −t t e −t t o −t t e 0 ∗ 0 +U ( U ψ ) −t t o (since U ψ is even and 0 ∗ 0 0 ∗ 0 U ψ is odd,( U ψ ) = U ψ ,( U ψ ) e o o e e o t t t t = U ψ , and we get) t o ∗ 0 ∗ 0 0 ∗ 0 =  ψ + U ( U ψ ) + U ( U ψ ) + ψ e o e e o o −t t −t t 4 The semiclassical limit on a star-graph with Kirchhoff… Page 13 of 43 45 1 1 ∗ 0 ∗ 0 0 ∗ 0 = ψ +  U ( U ψ ) + U ( U ψ ) . e o o e −t t −t t 2 4 On the other hand, − + ∗ 0 ∗ 0 ψ ψ U U =  U  U P −t t −t t ψ ψ ψ ψ ψ (Since P = P( + )/2 = ( − )/2 we get) e o e o ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) − U ( U ψ ) + U ( U ψ ) e e o −t t e −t t o −t t e 0 ∗ 0 −U ( U ψ ) −t t ∗ 0 ∗ 0 0 ∗ 0 =  ψ − U ( U ψ ) + U ( U ψ ) − ψ e o e o e o −t t −t t ∗ 0 ∗ 0 0 ∗ 0 ψ ψ =  −U ( U ) + U ( U ) . e o −t t o −t t e Hence 1 1 − − − + ∗ 0 ∗ 0 (U U − U U ) ψ = ψ +  U ( U ψ ) . o e −t t −t t −t t 2 2 Recall that U is the unitary group generated by the Dirichlet Laplacian on the half-line; its integral kernel is given by D 0 0 U (x , y) = U (x − y) − U (x + y) for x , y ∈ R . t t t Moreover, let U be the unitary group generated by the Neumann Laplacian on the half-line; its integral kernel is given by N 0 0 U (x , y) = U (x − y) + U (x + y) for x , y ∈ R . t t t Note that, for x ∈ R , ∗ 0 0 ψ ψ ( U )(x ) = dy U (x − y) (y) t o t o 0 0 D = dy U (x − y) − U (x + y) ψ(y) = (U ψ)(x), t t t similarly ∗ 0 0 0 N ψ ψ ψ ( U )(x ) = dy U (x − y) + U (x + y) (y) = (U )(x). t e t t t Hence, ∗ 0 ∗ 0 ∗ 0 D N D U ( U ψ ) =  U (U ψ) = U U ψ , o e e −t t −t t −t t 45 Page 14 of 43 C. Cacciapuoti et al. and 1 1 − − − + N D (U U − U U ) ψ = ψ + U U ψ . (2.15) −t −t t t −t t 2 2 A similar computation gives + + ∗ 0 ∗ 0 U U ψ =  U P  U P ψ −t t −t t ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) − U ( U ψ ) − U ( U ψ ) e e o e o e −t t −t t −t t 0 ∗ 0 +U ( U ) −t t o 1 1 ∗ 0 ∗ 0 0 ∗ 0 ψ ψ ψ = −  U ( U ) + U ( U ) , e o −t t o −t t e 2 4 and + − ∗ 0 ∗ 0 U U ψ =  U P  U  ψ −t t −t t ∗ 0 ∗ 0 0 ∗ 0 0 ∗ 0 ψ ψ ψ =  U ( U ) + U ( U ) − U ( U ) e e o −t t e −t t o −t t e 0 ∗ 0 −U ( U ψ ) o o −t t ∗ 0 ∗ 0 0 ∗ 0 =  U ( U ψ ) − U ( U ψ ) . o e e o −t t −t t Hence 1 1 1 1 + + + − ∗ 0 ∗ 0 N D (U U − U U ) ψ = ψ −  U ( U ψ ) = ψ − U U ψ . −t t −t t −t t o −t t 2 2 2 2 (2.16) N D To compute the wave operator we have to evaluate the limits s-lim U U ; t →±∞ −t t the latter give the wave operators  for the pair (Dirichlet Laplacian, Neumann ND Laplacian) on the half-line, which are computed in Proposition A.1 and equal ± F F . This, together with Eqs. (2.14), (2.15), and (2.16) concludes the proof of identity (2.13). Remark 2.3 Note that the wave operators do not depend on . Moreover the scattering operator is given by + ∗ − S = ( )  = S , (2.17) the same formula is written component-wisely in Eq. (1.7). The matrix S given here (v) (v) σ σ equals − , where is the scattering matrix at the vertex, as defined in [4, Def. 2.1.1.] (see also [4, Ex. 2.1.7, p. 41] and [16, Ex. 2.4]). The minus sign is due to the fact that as reference Hamiltonian we chose Dirichlet boundary conditions, instead of Neumann boundary conditions, see also [16]. The last identity in Eq. (2.17) can be derived by a simple computation starting from Eq. (2.13), recalling ∗ 2 that F and F are unitary operators, and noting that S = S, S = 1. s c The semiclassical limit on a star-graph with Kirchhoff… Page 15 of 43 45 3 The semiclassical theory 3.1 Classical dynamics on the half-line with elastic collision at the origin We start by recalling some basic definitions and results from [5]. Let X (q, p) = (p/m, 0) be the vector field associated with the free (classical) Hamiltonian of a particle of mass m and consider the differential operator 2 2 L : S (R ) → S (R ), Lf := −iX ·∇ f in the space of tempered distributions S (R ). We denote by p ∂ f 2 2 2 2 L : dom(L ) ⊆ L (R ) → L (R ), (L f )(q, p) =− i (q, p), 0 0 0 m ∂q 2 2 2 2 dom(L ) := f ∈ L (R ) Lf ∈ L (R ) , 2 0 −1 the maximal realization of L in L (R ). Posing R := (L − z) for z ∈ C\R one has im 0 imz(q−q )/p (R f )(q, p) = sgn(Imz) dq ((q − q ) pImz) e f (q , p). (3.1) | p| it L Moreover the action of the (free) unitary group e (t ∈ R)isgiven by pt it L e f (q, p) = f q + , p . For any f ∈ S(R ) we define the map ( f )(p) := f (0, p). For a comparison with the results in [5, Sec. 2], recall that the map can be equivalently defined as ˜ ˜ γ √ ( f )(p) = dk f (k, p) where f (k, p) is the Fourier transform of f (q, p) 2 π γ γ in the variable q.By[5, Lem. 2.1], the map extends to a bounded operator : 2 2 2 dom(L ) → L (R, | p| dp), where dom(L ) ⊂ L (R ) is endowed with the graph 0 0 0 2 2 2 norm. Hence, for any z ∈ C\R the operator R : L (R ) → L (R, | p| dp) is bounded, and so is its adjoint (in z ¯): 2 −1 2 2 0 ∗ G : L (R, | p| dp) → L (R ), G := ( R ) z z z ¯ 2 −1 2 (here L (R, | p| dp) and L (R, | p|dp) are considered as a dual couple with respect to the duality induced by the scalar product in L (R)). An explicit calculation gives im imzq/p (G u)(q, p) = (qpImz) sgn(Imz) e u(p). z θ | p| Next we consider the classical motion of a point particle of mass m on the whole real line, with elastic collision at the origin. The generator of the dynamics, denoted 45 Page 16 of 43 C. Cacciapuoti et al. by L , is obtained as a limiting case, for β →∞, of the operator L defined in [5]. ∞ β To this aim we set 2| p| ∞ 2 2 −1 ∞ : L (R, | p|dp) → L (R, | p| dp), ( u)(p) := isgn(Imz) u(p). z z In addition, let us consider the projector on even functions (here either (p) =| p| or −1 (p) =| p| ) 2 2 ρ ρ : L (R, dp) → L (R, dp), ( f )(p) := f (p) + f (−p) . ev ev By [5, Thm. 2.2], here employed with β →∞, the operator L is defined by 2 2 ∞ dom(L ) := f ∈ L (R ) f = f + G f , f ∈ dom(L ) , ∞ z z ev z z 0 (L − z) f = (L − z) f , ∞ 0 z ∞ −1 for all z ∈ C\R. The associated resolvent operator R := (L − z) (z ∈ C\R) can be expressed as follows, in terms of the free resolvent R and of the trace operator # $ ∞ 0 imzq/p 0 0 γ γ (R f )(q, p) = (R f )(q, p) − (qpImz) e ( R f )(p) + ( R f )(−p) . z z z z (3.2) More explicitly, we have im ∞ imz(q−q )/p (R f )(q, p) = sgn(Imz) dq (q − q ) pImz e f (q , p) | p| imz(q−q )/p − (qpImz) (−q pImz) e f (q , p) θ θ imz(q+q )/p + (q pImz) e f (q , −p) . (3.3) Correspondingly, let us recall that [5, Prop. 2.4] gives, for all t ∈ R, it L it L ∞ 0 e f (q, p) = e f (q, p) (3.4) |pt | it L it L 0 0 − θ(−tqp) θ −|q| e f (q, p) + e f (−q, −p) it L it L 0 0 (here, for a comparison with [5], we used e f (−· , −· ) (q, p) = e f (−q, −p) ). To proceed, let us introduce the lateral traces defined by # $ ∞ 0 0 0 γ γ γ γ (R f ) (p) := ( R f )(p) − θ(pImz) ( R f )(p) + ( R f )(−p) z z z z 0 0 γ γ = (−pImz)( R f )(p) − (pImz)( R f )(−p), (3.5) θ θ z z # $ ∞ 0 0 0 γ γ γ γ (R f ) (p) := ( R f )(p) − (−pImz) ( R f )(p) + ( R f )(−p) − θ z z z z 0 0 γ γ = (pImz)( R f )(p) − (−pImz)( R f )(−p) (3.6) θ θ z z The semiclassical limit on a star-graph with Kirchhoff… Page 17 of 43 45 (here we used the trivial identity (−s) = 1 − (s) ). Clearly, (R f ) are odd θ θ functions and, using again [5,Lem.2.1], : dom(L ) → L (R, | p|dp) (3.7) ± ∞ odd is a bounded operator. We remark that the action of can be understood as ∞ ∞ (R f ) (p) = lim ± (R f )(q, p). ± q→0 z z For the subsequent developments it is convenient to express the free resolvent R in terms of R . More precisely, we have the following explicit characterization. Lemma 3.1 For any z ∈ C\R and for any f ∈ L (R ), there holds 0 ∞ (R f )(q, p) = (R f )(q, p) z z imzq/p ∞ ∞ γ γ − θ(qpImz) sgn(q) e ( R f )(p) − ( R f )(p) . + − z z Proof From Eqs. (3.5) and (3.6) we readily infer that ∞ 0 γ γ (−pImz)( R f )(p) = (−pImz)( R f )(p), θ θ z z ∞ 0 γ γ (pImz)( R f )(p) = (pImz)( R f )(p). θ − θ z z The above relations imply, in turn, 0 ∞ ∞ γ γ γ ( R f )(p) = (−pImz)( R f )(p) + (pImz)( R f )(p), θ + θ − z z z and, since R f are odd functions, 0 ∞ ∞ γ γ γ ( R f )(−p) =− θ(pImz) R f (p) − θ(−pImz) R f (p). + − z z z Substituting the latter identities into Eq. (3.2) and noting that (qpImz) = (q) + (−q) (qpImz) = (q) (pImz) + (−q) (−pImz), θ θ θ θ θ θ θ θ and sgn(q) (qpImz) = (q) (pImz) − (−q) (−pImz), θ θ θ θ θ we obtain (R f )(q, p) 0 imzq/p ∞ = (R f )(q, p) − e (q) (pImz)( R f )(p) θ θ − z z − (q) (pImz)( R f )(p) θ θ + ∞ ∞ γ γ + (−q) (−pImz)( R f )(p) − (−q) (−pImz)( R f )(p) θ θ + θ θ − z z 45 Page 18 of 43 C. Cacciapuoti et al. 0 imzq/p ∞ ∞ γ γ = (R f )(q, p) + (qpImz) sgn(q) e ( R f )(p) − ( R f )(p) , θ + − z z z which suffices to infer the thesis. Similarly, for the unitary operator describing the dynamics we have 2 2 Lemma 3.2 For any t ∈ R and for any f ∈ L (R ), there holds it L it L e f (q, p) = e f (q, p) |pt | it L it L ∞ ∞ − (−tqp) −|q| e f (q, p)+ e f (−q,−p) . θ θ Proof First note that the identity in Eq. (3.4) entails it L it L ∞ 0 e f (−q, −p) = e f (−q, −p) |pt | it L it L 0 0 − θ(−tqp) θ −|q| e f (q, p)+ e f (−q,−p) . From the above relation and the previously cited equation, we derive it L it L ∞ ∞ e f (q, p) + e f (−q, −p) |pt | it L it L 0 0 = 1 − 2 θ(−tqp) θ −|q| e f (q, p) + e f (−q, −p) , which in turn implies |pt | it L it L ∞ ∞ (−tqp) −|q| e f (q, p) + e f (−q, −p) θ θ |pt | it L it L 0 0 =− (−tqp) −|q| e f (q, p) + e f (−q, −p) . θ θ The thesis follows upon substitution of the above identity into Eq. (3.4). Let us now consider the natural decomposition 2 2 2 L (R ) ≡ L (R ∪ R )×R, dqdp − + 2 2 2 = L (R , dq) ⊕ L (R , dq) ⊗ L (R, dp) − + 2 2 2 2 = L (R , dq) ⊗ L (R, dp) ⊕ L (R , dq) ⊗ L (R, dp) − + 2 2 = L (R × R) ⊕ L (R × R), (3.8) − + All the equalities in Eq. (3.8) must be understood as isomorphisms of Hilbert spaces (see, e.g., [13,p. 85]). The semiclassical limit on a star-graph with Kirchhoff… Page 19 of 43 45 2 2 2 and notice that both the subspaces L (R ×R) ≡ L (R ×R)⊕{0} and L (R ×R) ≡ − − + 2 ∞ {0}⊕ L (R × R) are left invariant by the resolvent R , this is evident from Eq. (3.3). Taking this into account, we introduce the bounded operator D 2 2 D ∞ R : L (R × R) → L (R × R), (R f )(q, p) := R (0 ⊕ f ) (q, p). (3.9) + + z z z By direct computations, from Eq. (3.3) (here employed with q > 0) we get R f (q, p) im imz(q−q )/p = sgn(Imz) dq θ (q − q )pImz e f (q , p) | p| imz(q+q )/p − (pImz) e f (q , −p) im imzq/p −imzq /p = sgn(Imz) e (pImz) dq e f (q , p) | p| imzq /p − dq e f (q , −p) −imzq /p + θ(−pImz) dq e f (q , p) . 2 D We denote by L the self-adjoint operator in L (R × R) having R as resolvent, D + so that dom(L ) = f ∈ L (R × R) (0 ⊕ f ) ∈ dom(L ) , L f := L (0 ⊕ f ). D + ∞ D ∞ (3.10) For all (q, p) ∈ R ×R, t ∈ R and f ∈ L (R ×R), from the above definition and from + + Eq. (3.4) we get it L it L D ∞ e f (q, p) = e (0 ⊕ f ) (q, p) pt it L = q + e (0 ⊕ f ) (q, p) pt it L − − q − e (0 ⊕ f ) (−q, −p), (3.11) which describes the motion of a classical particle on the half-line R with elastic collision at q = 0. Let us also mention that, in view of the basic identity pt it L it L 0 0 e (0 ⊕ f ) (q, p) = θ q + e (0 ⊕ f ) (q, p), (3.12) |pt | pt pt Note that for q > 0we have θ(−tqp) θ −|q| = θ(−tp) θ − − q = θ −q − = m m m pt 1 − q + . m 45 Page 20 of 43 C. Cacciapuoti et al. the above relation (3.11) is equivalent to it L it L it L D 0 0 e f (q, p) = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, −p). (3.13) Another equivalent (and more explicit) formula for the action of the unitary group it L e is the one given in Eq. (1.8). Finally, from Lemmata 3.1 and 3.2 (here employed with q > 0) we derive, respec- tively, 0 D imzq/p D R (0 ⊕ f ) (q, p) = (R f )(q, p) − (pImz) e ( R f )(p), θ + z z z pt it L it L 0 D e (0 ⊕ f ) (q, p) = θ q + e f (q, p) pt it L − −q − e f (−q, −p). (3.14) 3.2 Classical dynamics on the graph with total reflection at the vertex n 2 Let us now consider the “classical” Hilbert space ⊕ L (R × R) and indicate any =1 of its elements with the vector notation ⎛ ⎞ f (q, p) ⎜ ⎟ n 2 f ∈⊕ L (R × R), f (q, p) ≡ . + ⎝ ⎠ =1 . f (q, p) Let L be defined according to Eq. (3.10), and consider the classical dynamics on the star-graph G with total elastic collision in the vertex; this is described by the self-adjoint operator ⊕ n L := ⊕ L . D =1 The associated resolvent and time evolution operators are respectively given by D⊕ −1 n D R := (L − z) =⊕ R , z z D =1 it L n it L e =⊕ e . =1 Explicitly, for  = 1,..., n and t ∈ R,fromEq. (1.8) we derive pt pt f q + , p if q + > 0 , m m it L e f (q, p) = pt pt − f − q − , −p if q + < 0 . m m The semiclassical limit on a star-graph with Kirchhoff… Page 21 of 43 45 3.3 Singular perturbations of the classical dynamics on the graph Let us consider the restriction to dom(L ) of the trace map introduced in Eq. (3.7); D + this defines a bounded operator : dom(L ) → L (R, | p|dp). + D odd We use the above map to define a trace operator on the graph: n n n 2 γ γ := ⊕ :⊕ dom(L ) →⊕ L (R, | p|dp). + D =1 =1 =1 odd In what follows we use the technique developed by one of us in [20,21] to char- ⊕ ⊕ acterize all the self-adjoint extensions of the symmetric operator L  ker( ) (see Theorem 3.3 below). Among those we select the one that turns out to be useful to study the semiclassical limit of exp(−iH t /),see Remark 3.4. To proceed, we introduce the operator ⊕ ⊕ D⊕ n + n 2 n 2 ˘ ˘ G := R ≡⊕ G :⊕ L (R × R) →⊕ L (R, | p|dp), z + z =1 z  =1 odd + D G := R , z z and its adjoint with complex conjugate parameter, ⊕ ⊕ n + n 2 n 2 G := G ≡⊕ G :⊕ L (R, | p|dp) →⊕ L (R × R), z =1 z  odd =1 z ¯ + + G := G . z ¯ Note that by Eq. (3.1) one has im 0 −imzq /p R (0 ⊕ f ) (p) = sgn(Imz) dq θ(−pImz) e f (q , p). | p| From the latter identity, together with Eqs. (3.9) and (3.5), we derive + ∞ (G f )(p) = R (0 ⊕ f ) (p) z z 0 0 γ γ = (−pImz) R (0 ⊕ f ) (p) − (pImz) R (0 ⊕ f ) (−p), θ θ z z im −imzq /p =sgn(Imz) dq (−pImz) e f (q , p) | p| imzq /p − (pImz) e f (q , −p) . (3.15) 2 2 In view of the latter expression, for all f ∈ L (R × R) and any φ ∈ L (R, | p|dp) odd we have + + φ φ dqdp (G )(q, p) f (q, p) = dp (p)(G f )(p) z ¯ R ×R R + 45 Page 22 of 43 C. Cacciapuoti et al. im −imzq ¯ /p = dp φ(p) sgn(Imz ¯) dq (−pImz ¯) e f (q, p) | p| R 0 imzq ¯ /p − (pImz ¯) e f (q, −p) im −imzq ¯ /p φ φ = dqdp (p) − (−p) θ(pImz) sgn(Imz) − e f (q, p) | p| R ×R 2im imzq/p = dqdp (pImz) sgn(Imz) e φ(p) f (q, p), | p| R ×R which proves that im + imzq/p (G φ)(q, p) = 2 g (q, p) φ(p), g (q, p) := (pImz) sgn(Imz) e . z z | p| (3.16) On account of the identities im (g − g ) (p) = sgn(Imz) − sgn(Imw) , + z w 2 | p| which can be easily checked by a direct calculation (see also [5,p.7]),and + + + ∗ + (G − G ) = (z − w) (G ) G , z w z w ¯ which is consequence of the first resolvent identity (see [20, Lem. 2.1], paying attention to the different sign convention in the definition of the resolvent), we have that the linear map + 2 −1 2 dom(M ) := L (R, | p| dp) ∩ L (R, | p|dp), z odd odd + + 2 −1 2 M : dom(M ) ⊂ L (R, | p| dp) → L (R, | p|dp), z z odd odd im + ∞ ∞ φ φ (M )(p) := 2 m (p) (p), m (p) := − sgn(Imz) , (3.17) z z z 2| p| satisfies the identities + ∗ + + + + ∗ + (M ) = M , M − M = (w − z)(G ) G . z z w z z ¯ w ¯ Hence, setting ⊕ n + n 2 −1 n 2 M :⊕ dom(M ) ⊂⊕ L (R, | p| dp) →⊕ L (R, | p|dp), z =1 z =1 odd =1 odd ⊕ n + M := ⊕ M , z =1 z one gets the identities ⊕ ∗ ⊕ ⊕ ⊕ ⊕ ∗ ⊕ (M ) = M , M − M = (w − z)(G ) G . z z ¯ z w w ¯ z The semiclassical limit on a star-graph with Kirchhoff… Page 23 of 43 45 n n To proceed, let us consider any orthogonal projector : C → C and any self-adjoint n n operator B : C → C , represented by the matrices with components ( ) and (B) respectively. By a slight abuse of notation we use the same symbols to denote the corresponding operators on vector valued functions; e.g., for f =⊕ f ∈ =1 n 2 n 2 ⊕ L (R × R), one has f ∈⊕ L (R × R) with components ( = 1,..., n) + + =1 =1 f ) = ( ) f , =1 n n 2 n n or, for ⊕ φ ∈⊕ L (R, dp), one has (⊕ φ ) := ⊕ =1 =1 odd =1 =1 ) , and similarly for B. Then, by [20, Thm. 2.1] here employed j =1 with τ := , we obtain the following ⊕ ⊕ Theorem 3.3 Let z ∈ C\R. Assume that M = M and B = B . Then, the z z linear bounded operator −1 ,B D⊕ ⊕ ⊕ ⊕ R := R + G B + M G (3.18) z z z z z is the resolvent of a self-adjoint extension L of the densely defined, closed sym- ,B ⊕ ⊕ metric operator L  ker( ). Such an extension acts on its domain n 2 dom(L ) := f ∈⊕ L (R × R) ,B =1 −1 ⊕ ⊕ ⊕ ⊕ f = f + G B + M f , f ∈dom L z z z z z + D according to L − z f = L − z f . (3.19) ,B Remark 3.4 We use the notation L (where K stands for Kirchhoff) to denote the self-adjoint extension corresponding to the choices ⎛ ⎞ 11 ··· 1 ⎜ ⎟ 11 ··· 1 ⎜ ⎟ B = 0, = . (3.20) ⎜ . . .⎟ . . . . n ⎝ ⎠ . . . 11 ··· 1 K −1 We denote the associated resolvent operator with R := (L − z) . In the sequel, we proceed to determine the unitary evolution associated with the above choices by means of arguments analogous to those described in the proof of [5, Prop. 2.4]. 45 Page 24 of 43 C. Cacciapuoti et al. Proposition 3.5 For all f ∈⊕ L (R × R) and for all t ∈ R there holds =1 ⎛   ⎞ ⎛   ⎞ it L it L 0 0 e (0⊕ f ) (q, p) e (0⊕ f ) (−q, −p) 1 1 ⎜ . ⎟ ⎜ . ⎟ it L . . (e f )(q, p) = − S (3.21) ⎝ ⎠ ⎝ ⎠ . . it L it L 0 0 e (0⊕ f ) (q, p) e (0⊕ f ) (−q, −p) n n where S := 1 − 2 was already defined in relation with the quantum scattering operator, see Eq. (2.8). Proof Throughout the whole proof we work component-wisely, denoting with  ∈ {1, ..., n} a fixed index. Let us first remark that the resolvent (3.18), with B, as in Eq. (3.20), acts on any element f ∈⊕ L (R × R) according to =1 n n K D D R f (q, p) = (R f )(q, p) + 2 g (q, p) ( ) ( R f )(p) z j j + j z z z 2 m (p) j =1 j =1 g (q, p) 1 D D = (R f )(q, p) + ( R f )(p). + j z z m (p) n j =1 From the above relation we derive the following, recalling the explicit expressions for g (q, p) and m (p) given in Eqs. (3.16) and (3.17), as well as Eq. (3.15)for D + R f = G f : z z K D R f (q, p) = (R f )(q, p) z z 2im imz(q+q )/p + (pImz) sgn(Imz) dq e f (q , −p). n | p| j =1 −it L We now proceed to compute the unitary operator e (t ∈ R) by inverse Laplace transform, using the above representation for the resolvent R . Let us first assume t > 0; then, for any c > 0 and f ∈ dom(L ), we get (see [10, Ch. III, Cor. 5.15]) r +ic −it L −izt K e f = lim dz e R f r →∞ 2 π i −r +ic ct r −itk K = lim dk e R f . k+ic π r →∞ 2 i −r On the one hand, recalling Eq. (3.13)wehave ct r −itk D −it L lim dk e (R f )(q, p) = (e f )(q, p) k+ic r →∞ 2 π i −r −it L −it L 0 0 = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, −p). The semiclassical limit on a star-graph with Kirchhoff… Page 25 of 43 45 On the other hand, noting that pIm(k+ic) = (pc) = (p) and sgn Im(k+ic) = θ θ θ sgn(c) =+1for c > 0, by computations similar to those reported in the proof of [5, Prop. 2.4] we get ct r e 2im −itk lim dk e pIm(k +ic) sgn Im(k +ic) r →∞ 2 π i n| p| −r im(k+ic)(q+q )/p dq e f (q , −p) j =1 c(t −mq/p) r 2m e i (mq/p−t )k = (p) lim dk e r →∞ n| p| 2 π −r j =1 im(k+ic)q /p dq e (q ) f (q , −p) θ j 2m c(t −mq/p) = (p) e n| p| −1 F F ( · ) f ( · , −p) − m(∗+ ic)/p (mq/p − t ) j =1 2 pt pt = (p) − q + f −q + , −p θ θ n m m j =1 −it L = e (0 ⊕ f ) (−q, −p). j =1 Summing up, the above relations imply −it L −it L −it L K 0 0 e f = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, −p) −it L + e (0 ⊕ f ) (−q, −p). (3.22) j =1 For t < 0 one can perform similar computations, starting from the following identity where c > 0: r −ic −it L −izt K e f =− lim dz e R f π r →∞ 2 i −r −ic −ct r −itk K =− lim dk e R f . k−ic r →∞ 2 π i −r Especially, recall the following basic identity regarding the unitary Fourier transform F and its inverse −1 F : cq −1 F Fh a(∗+ ic) (q) = h(q/a) fora ∈ R\{0}, c > 0, q ∈ R , |a| c· e 2 which holds true whenever h(·/a) ∈ L (R). In addition, keep in mind the relation written in Eq. (3.12). |a| 45 Page 26 of 43 C. Cacciapuoti et al. We omit the related details for brevity. In the end, one obtains exactly the same expres- sion as in Eq. (3.22), which with the trivial replacement t →−t proves Eq. (3.21). it L Remark 3.6 By Eq. (3.12) we infer that the action of the unitary group e is explicitly given, component-wisely, by Eq. (1.9). Remark 3.7 Recalling the explicit form of (see Eq. (3.20)) we obtain ⎛ ⎞ ⎛ ⎞ n − 2 −2 ··· −2 n − 1 −1 ··· −1 ⎜ ⎟ ⎜ ⎟ −2 n − 2 ··· −2 −1 n − 1 ··· −1 1 1 ⎜ ⎟ ⎜ ⎟ S = , 1 − = . (3.23) ⎜ ⎟ ⎜ ⎟ . . . . . . . . . . . . . . . . ⎝ ⎠ ⎝ ⎠ n n . . . . . . . . −2 −2 ··· n − 2 −1 −1 ··· n − 1 In particular, for a star-graph with three edges (n = 3) we have ⎛ ⎞ 1 −2 −2 ⎝ ⎠ S = −21 −2 , −2 −21 whence S =− M with respect to the notation used in [1]. 3.4 The semiclassical wave operators and scattering operator Let us consider the wave operators and the corresponding scattering operator on ⊕ L (R ×R), respectively defined by Eqs. (1.10) and (1.11). The following propo- =1 sition provides explicit expressions for these operators. Proposition 3.8 The limits in Eq. (1.10) exist point-wisely for any ξ ≡ (q, p) ∈ R × 2 n 2 (R\{0}) and in L (R × R) for any f ∈⊕ L (R × R); moreover, there holds + + =1 # $ # $ f (q, p) = 1 − (∓p) 2 f (q, p) = (±p) 1 + (∓p) S f (q, p). θ θ θ cl (3.24) Furthermore, the scattering operator is given by S = 1 − 2 = S . (3.25) cl Proof First of all let us point out that for all t ∈ R and any  ∈{1, ..., n},Eqs.(3.13) and (3.21) give, respectively, it L it L it L D 0 0 e f (q, p) = e (0 ⊕ f ) (q, p) − e (0 ⊕ f ) (−q, − p) pt pt = (0 ⊕ f ) q + , p − (0 ⊕ f ) −q − , − p m m pt pt pt pt = θ q + f q + , p − θ −q − f −q − , − p ; m m m m −it L −it L −it L K 0 0 (e f ) (q, p) = e (0 ⊕ f ) (q, p) + 2 −1 e (0 ⊕ f ) (−q, − p) j =1 The semiclassical limit on a star-graph with Kirchhoff… Page 27 of 43 45 pt pt = (0 ⊕ f ) q − , p + 2 −1 (0 ⊕ f ) −q + , − p m m j =1 pt pt = q − f q − , p m m pt pt + 2 −1 θ −q + f −q + , − p . m m j =1 In view of the above relations, recalling that q > 0, by direct computations we obtain it L −it L it L −it L K D K e e f (q, p) = e (e f ) (q, p) pt pt −it L = q + (e f ) q + , p m m pt pt −it L − −q − (e f ) −q − , −p m m pt pt = q + f (q, p) − −q − 2 −1 f (q, p) θ θ m m j =1 pt = f (q, p) − θ −q − 2 f (q, p), j =1 which gives ⊕ pt it L −it L e e f (q, p) = f (q, p) − −q − 2 f (q, p). pt Then Eq. (3.24) follows noting that −q − → (∓p) for t →±∞. θ θ Next, since is symmetric, by elementary arguments we get # $ ± ∗ ( ) f (q, p) = 1 − (∓p) 2 f (q, p). cl On account of the above identity we obtain # $ + ∗ − − S f (q, p) = ( )  f (q, p) = 1 − θ(−p) 2 f (q, p) cl cl cl cl # $# $ # $ = 1 − θ(−p) 2 1 − θ(p) 2 f (q, p) = 1 − 2 f (q, p), which proves Eq. (3.25). 4 Comparison of the semiclassical and quantum theories Recall the definition of coherent states in L (R) given in Eqs. (1.1), and (1.2) describ- ing their free evolution. 45 Page 28 of 43 C. Cacciapuoti et al. −it L Additionally, notice that the action of the operators e ,  , and S can be cl cl extended in a natural way to ⊕ L (R × R). =1 For later reference let us point out the following auxiliary result, regarding the functions ψ defined in Eq. (1.13), and the operators U defined in Eq. (2.3). σ,ξ t Lemma 4.1 For all ξ = (q, p) ∈ R × R and for any t ∈ R there holds i 1 ± A 4σ U ψ − e ψ ≤ √ e . σ ξ σ ξ 2 t , ,∓ 0 t t L (R ) Proof Let us first remark that, on account of the considerations reported in Remark 2.1, ψ =  ψ and for any x > 0 we have the chain of identities σ ,ξ σ ,ξ 0 0 ± 0  0 U ψ (x ) = U ψ (∓x ) = U ψ (∓x ) + E (∓x ) t σ ,ξ t σ ,ξ t σ ,ξ t 0 0 0 = e (∓x ) + E (∓x ) σ ξ t t t i i A   A t t = e ψ (x ) + E (∓x ) = e ψ (x ) + E (∓x), σ ,∓ξ t σ ,∓ξ t t t t where we put E (x ) := − U (1 − ) ψ (x ) for convenience of notation and used t t σ ,ξ the trivial identity ψ (−x ) = ψ (x ). σ,ξ σ,− ξ On the other hand, via an explicit calculation involving the trivial inequality 2 2 2 η η − (a+b) − (a +b ) e ≤ e for , a, b ≥ 0 we obtain 2 2 2 2 E (∓· ) ≤ E = U (1 − ) ψ = (1 − ) ψ 2 2 θ 2 θ 2 t t t σ ,ξ σ ,ξ L (R ) L (R) 0 L (R) 0 L (R) 2 2 (x +q) q 0 ∞ − − 1 2 1 2 2 2σ 2σ 0 0 = dx |ψ (x )| = √ dx e ≤ e , (4.1) σ ,ξ 2 π  σ −∞ 0 which proves the thesis in view of the previous arguments. To proceed let us point out the forthcoming lemma which characterizes a large class of functions satisfying the condition in Definition 1.1. Lemma 4.2 Let > 0, ξ = (q, p) ∈ R × R and consider a family of functions ∞ ∞ χ η ∈ L (R ), uniformly bounded in L (R ) with respect to , q and such that q, + + η(x ) = 1 for |x − q| < q . (4.2) q, Then, the functions  ∈ L (R ) defined by σ,ξ (x ) := (x ) ψ (x)(x > 0) (4.3) q,η σ,ξ ,ξ q, σ,ξ L (R ) fulfill the condition (1.14) with ε < min{1/4, /8} and C depending only on ∞ . q, L (R ) + The semiclassical limit on a star-graph with Kirchhoff… Page 29 of 43 45 Proof Let us first remark that the states  defined in Eq. (4.3) have unit norm in ,ξ L (R ) by construction; moreover, again from Eq. (4.3) it follows that ψ = + q, σ,ξ . Taking these facts into account, we have η 2 q, σ,ξ L (R ) σ + ,ξ χ χ ψ ψ ψ − ≤  − η + ( η − 1) 2 q, 2 q, 2 σ ξ σ,ξ σ ξ σ,ξ σ,ξ , L (R ) , L (R ) L (R ) + + + = 1 − q, 2 σ,ξ L (R ) σ,ξ + L (R ) + ( η − 1) ψ q, 2 σ,ξ L (R ) χ χ = 1 − + ( − 1) ψ . η 2 η q, q, 2 σ,ξ L (R ) σ,ξ + L (R ) On one hand, recalling the definition (1.13)of ψ and that = 1, using σ,ξ σ ξ L (R) 2 2 2 the basic inequality (a − b) ≤|a − b | for a, b > 0 we get 2 2 χ χ 1 − η 2 η q, 2 q, 2 σ,ξ L (R ) σ,ξ σ,ξ + L (R) L (R ) = dx |ψ (x )| − dx (x ) ψ (x ) q, σ,ξ σ,ξ R 0 2 2 ≤ dx 1 − (x ) (x ) ψ (x ) θ η q, σ,ξ 2 2 2 ≤ dx 1 − (x ) ψ (x ) + dx (x ) 1 − η(x ) ψ (x ) . θ θ q, σ σ ,ξ ,ξ R R Recalling the hypothesis (4.2), the explicit expression (1.1)for ψ , and that we are σ,ξ assuming q > 0, from the above results we derive 2 2 ψ ψ 1 − 2 ≤ dx (x ) q, σ,ξ L (R ) σ ξ + , −∞ 2 2 + dx 1 − η(x ) ψ (x ) q, ,ξ R ∩{|x −q| > q} (x +q) 1 − 2| σ | ≤ √ dx e 2 π  | σ | (x −q) 2| σ | + 1 + ∞ dx e q, L (R ) {|x −q| > q} q ∞ − − 2 2 2| σ | 2| σ | ≤ e dx e 2 π  | σ | 0 2 2 2 q (x −q) − − 2 2 4| σ | 4| σ | + 1 + ∞ e dx e q, L (R ) 2 2 2 q q − − 2 2 2| σ | 4| σ | ≤ e + 2 1 + ∞ e . q, L (R ) 2 45 Page 30 of 43 C. Cacciapuoti et al. On the other hand, by arguments similar to those employed above we get 2 2 2 χ χ ψ ψ ( η − 1) = dx η(x ) − 1 (x ) q, 2 q, σ,ξ σ,ξ L (R ) (x −q) 1 − 2| | χ ∞ ≤ 1 + √ dx e q, L (R ) 2 π  | σ | η {|x −q| > q} 2 2 η q 4| σ | ≤ 2 1 + e . q, L (R ) √ √ Summing up, the previous results and the basic relation a + b ≤ a + b for a, b > 0imply 2 2 2 q q 1 − − 2 5/4 2 σ σ 4| | 8| | χ ∞ − ψ ≤ √ e + 2 1 + 2 q, L (R ) σ σ,ξ + ,ξ L (R ) 1 − ε 5/4 2 | σ | ≤ √ + 2 1 + e , q, L (R ) which suffices to infer the thesis on account of the uniform boundedness of η. q, Example 4.3 For ∈ (0, 1], consider the sharp cut-off functions 0if x ≤ (1 − ) q , (x ) = q, 1if x >(1 − ) q , which clearly satisfy the hypothesis of Lemma 4.2. The corresponding elements ∈ L (R ) defined according to Eq. (4.3) consist of normalized truncations σ ξ of the coherent state ψ and fulfill the condition (1.14) as a consequence. σ,ξ η χ It is worth noting that for = 1wehave ≡ 1on R , so that the associated q,η + function  is just the re-normalization of the bare truncation ψ introduced in Eq. σ,ξ σ,ξ (1.13), i.e., = ψ 2 . σ σ L (R ) σ,ξ ,ξ ,ξ + Example 4.4 For ∈ (0, 1/2), consider the smooth functions on R such that 0for |x − q| >(1 − ) q , χ χ (x ) = (x ) ≤ 1 . η η q, q, 1for |x − q| < q , Again, the assumptions of Lemma 4.2 are certainly verified and the related functions have compact support in R , besides satisfying the bound (1.14). ,ξ The semiclassical limit on a star-graph with Kirchhoff… Page 31 of 43 45 In addition to states fulfilling the requirements of Definitions 1.1 and 1.3, our arguments will often involve the unnormalized element ⎛ ⎞ σ ,ξ ⎜ ⎟ ⎜ ⎟ ∈ L (G),  ≡ , (4.4) ⎜ ⎟ σ ,ξ σ ,ξ . 0 0 ⎝ . ⎠ along with its classical counterpart ⎛ ⎞ ,x ⎜ ⎟ ⎜ ⎟ ∈ L (G),  ≡ ⎜ ⎟ , (4.5) σ σ ,x ,x . 0 0 ⎝ . ⎠ with defined as in Eq. (1.13) and σ ,ξ φ (ξ) := ψ (x). σ ,x σ ,ξ 0 0 4.1 Comparing the dynamics. Proof of Theorem 1.4 Let  and  (ξ) be, respectively, as in Eqs. (4.4) and (4.5), and note that from σ ,ξ σ ,(·) 0 0 the triangular inequality it follows t i −i H  A it L K t K e  − e e  (ξ) σ σ ,ξ ,(·) 0 t L (G) t t −i H  −i H K K ≤ e  − e σ ,ξ σ ,ξ 0 0 L (G) t i −i H  A it L K t K + e  − e e  (ξ) σ σ ,ξ ,(·) 0 t L (G) i i A it L  A it L t K t K + e e  (ξ) − e e  (ξ) . (4.6) σ ,(·) σ ,(·) t t L (G) −i H Regarding the first term on the right-hand side of Eq. (4.6), by the unitarity of e and the condition (1.14) we infer t t −i H  −i H K K e  − e  =  − 2 2 σ ,ξ σ ,ξ σ ,ξ σ ,ξ 0 0 L (G) 0 0 L (G) − ε =  − ψ ≤ C e . 2 0 σ ,ξ σ ,ξ 0 0 L (R ) As for the second term in Eq. (4.6), note that Eqs. (2.9) and (3.21)give t i −i H  A it L K t K e  − e e  (ξ) σ ,ξ σ ,(·) 0 t 45 Page 32 of 43 C. Cacciapuoti et al. ⎛   ⎞ − A it L t 0 U ψ − e e (0⊕φ ) (ξ) t σ ,ξ σ ,(·) 0 t ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + A it L U ψ − e e (0⊕φ ) (− ξ) t σ σ ,ξ ,(·) 0 t ⎜ ⎟ ⎜ ⎟ − S . ⎜ ⎟ ⎝ . ⎠ it L Since e (0⊕φ ) (± ξ) = φ (±ξ ) = ψ (x ) for x ∈ R , from the above σ ,x σ ,x t σ ,±ξ t t t identity and from Lemma 4.1 we deduce t i −i H  A it L K t K e  − e e  (ξ) σ ξ σ , ,(·) L (G) 0 t − A ≤ 2 U ψ − e ψ σ σ 2 t ,ξ ,ξ 0 t t L (R ) 2 2 2 + A 2σ + 2 |S | U ψ − e ψ ≤ 2 e . σ σ 2 t ,ξ ,−ξ 0 t t L (R ) =1 Let us finally consider the third term in Eq. (4.6). Recalling again the identity (3.21), we obtain it L  it L K K e  (ξ) − e  (ξ) σ σ ,(·) ,(·) t t ⎛     ⎞ it L it L 0  0 e (0⊕φ ) (ξ) − e (0⊕ ) (ξ) ,(·) σ ,(·) t t ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛     ⎞ it L it L 0  0 e (0⊕φ ) (− ξ) − e (0⊕ ) (− ξ) σ ,(·) σ ,(·) ⎜ ⎟ ⎜ ⎟ − S⎜ ⎟ . ⎝ . ⎠ From the above identity, by arguments similar to those employed previously we get it L  it L K K e  (ξ) − e  (ξ) σ ,(·) σ ,(·) t t L (G) Note also that, on account of Eq. (3.23), we have n n 2 2 n − 2 2 2 2 2 1 + |S | = 1 +|S | + |S | = 1 + + (n − 1) = 2 . 1 11 1 n n =1 =2 The semiclassical limit on a star-graph with Kirchhoff… Page 33 of 43 45 ≤ 2 ψ − σ ,ξ σ ,ξ t t L (R ) t t + − 2 ε 2  2 2 |σ | + 2 |S | ψ −  ≤ 4C e . 1 2 σ ,−ξ σ ,−ξ 0 t t L (R ) t t + =1 Summing up, the above bounds imply the thesis. 4.2 Comparing the wave and scattering operators. Proof of Theorem 1.6 T T Note the identity S (1, 0,..., 0) = (1 − 2/n, −2/n,... , −2/n) and recall the expression of  given in Eq. (2.13). Then, by simple computations we get ( = 1,..., n) 1 − (1 ∓ F F )  if  = 1 , c σ ,ξ (  ) = ,ξ − (1 ∓ F F ) if  = 1 . c σ ,ξ Additionally, recalling the expression of  given in Eq. (3.24), we get cl 1 − (∓p)  (ξ) if  = 1 , ⎨ θ σ ,x (  ) (ξ) = cl σ ,x − (∓p)  (ξ) if  = 1 . σ ,x In view of these results together with the identity (1.15) we derive ±   ∗ − (  )(ξ) = √ 1 ∓ F F − 2 (∓ p) 2 s 2 σ ,ξ cl σ ,(·) c σ ,ξ 0 0 L (G) 0 L (R ) = √ (1 − 2 θ(∓ p))F ∓ F  . c s 2 σ ,ξ 0 L (R ) By the bound (1 − 2 (∓p))F ∓ F ≤|1 − 2 (∓p)| ≤ 2 and by Eq. θ c s θ c s (1.14), we infer (1 − 2 (∓p))F ∓ F c s 2 σ ,ξ L (R ) 0 + − ε ≤ 2C e + (1 − 2 (∓p))F ∓ F ψ . 0 c s 2 σ ,ξ 0 L (R ) In what follows we prove the following upper bound which concludes the proof of the theorem 2 2 σ p √ − − 4σ (1 − 2 (∓p))F ∓ F ψ ≤ 2 e + e . (4.7) θ c s 2 σ ,ξ 0 L (R ) + 45 Page 34 of 43 C. Cacciapuoti et al. We start with the identity (1 − 2 (∓p))F ∓ F ψ θ c s σ 2 ,ξ 0 L (R ) ∞ ∞ = dk dx 1 − 2 (∓p) cos(kx ) ± i sin(kx ) ψ (x ) . σ ,ξ 0 0 Considering separately the cases p > 0 and p < 0 for the two possible choices of the signs, it is easy to convince oneself that (1 − 2 (∓p))F ∓ F ψ θ c s σ 2 ,ξ 0 L (R ) ∞ ∞ ⎪ ikx dk dx e ψ (x ) if p > 0 , σ ,ξ 0 0 ∞ ∞ ⎪ −ikx dk dx e ψ (x ) if p < 0 . σ ,ξ 0 0 Recall that the Fourier transform of ψ is given by σ ,ξ 1/4 1 √ 2 2 2 −ikx  −σ (k−p/) −ikq Fψ (k) := √ dx e ψ (x ) = σ e . σ ,ξ σ ,ξ 0 0 2 R Let us assume p > 0, we have the chain of identities/inequalities 1/2 ∞ ∞ ikx dk dx e ψ (x ) = 2 F ψ (−·) σ ,ξ σ ,ξ 0 0 L (R ) 0 0 ≤ 2 F (−·) ,ξ L (R ) + 2 F(1 − )ψ (−·) . σ ,ξ L (R ) 0 + Reasoning like for the bound in Eq. (4.1), we obtain 1/2 2 2 2 p 1 0 Fψ (−·) = dk Fψ (−k) ≤ √ e σ ,ξ σ ,ξ 0 L (R ) 0 and F(1 − )ψ (−·) ≤ F(1 − )ψ θ θ 2 2 σ ,ξ σ ,ξ 0 L (R ) 0 L (R) = (1 − )ψ σ ,ξ 0 L (R) 1/2 q 2 − 4σ = dx ψ (x ) ≤ √ e , ,ξ −∞ which conclude the proof of the bound (4.7)for p > 0. The proof of the bound for p < 0 is identical and we omit it. The semiclassical limit on a star-graph with Kirchhoff… Page 35 of 43 45 Identity (1.18) follows immediately from Eqs. (1.7) and (1.12). 5 Final remarks 5.1 A comparison with different approaches to the definition of a classical dynamics on the graph Our approach to the semiclassical limit was inspired by Hagedorn’s work [12]. In general, a coherent state (on the real-line) is the wave function ψ : R → C defined as σ ˘ 1 i ψ ψ σ σ ˘ (x ) = ( , , q, p; x ) := √ exp − (x − q) + p(x − q) , 1/4 (2 π ) σ 4 2 −1 −2 σ σ ˘ σσ ˘ σ with (p, q) ∈ R and , ∈ C\{0} such that Re( ) =| | > 0. In his seminal paper [12], Hagedorn provides the semiclassical evolution of a coher- ent state in the presence of a regular (at least C (R)) interaction potential V.By one of the main results in [12], the quantum evolution of the coherent state ψ is close (with respect to the L (R)-norm and for  small enough) to the wave function ψ σ σ ˘ x → e ( , , q , p ; x ), where the pair (q , p ) is the solution of the Hamilton t t t t t t equations ⎪ q ˙ = p , t t p ˙ =−V (q ), t t (q , p ) = (q, p), 0 0 the pair (σ , σ ˘ ) is given by t t ∂q i ∂q ∂ p ∂ p t t t t σ = σ + σ ˘ , σ ˘ =−2i σ + σ ˘ t t ∂q 2 ∂ p ∂q ∂ p and A is the classical action t 2 A = ds − V (q ) . t s 2m In our notation, one can associate with the quantum state ψ the phase space function (q, p) → (φ (σ, σ ˘ , x ))(q, p) := ψ (σ, σ ˘ , q, p; x ). By this correspondence, one gets it L ψ (σ , σ ˘ , q , p ; x ) = e φ (σ , σ ˘ , x ) (q, p), where L denotes the Liouville t t t t t t V operator associated with the vector field of the classical Hamiltonian + V (q);this 2m it L is analogous to e  (ξ), ξ ≡ (q, p). σ ,x We remark that, unlike the case of a quantum particle in the presence of a regular potential, in general there is no trajectory of a classical particle which describes the 45 Page 36 of 43 C. Cacciapuoti et al. −i H semiclassical limit of a quantum evolution of the form e  . As a conse- σ ,ξ quence, the semiclassical dynamics is not described by the Hamilton equations. One way to overcome this difficulty is to assign a probability to every possible path on the graph. Typically, the probability of a certain path is postulated, and given in terms of the square modulus of the quantum transition (or stability) amplitudes (see, e.g., [3, Sec. II.A] or [4, Sec. 6.1]). For a star-graph the latter coincide with the elements of the (vertex) scattering matrix, defined for generic boundary conditions, e.g., in [16, Thm. 2.1] or [4, Lem. 2.1.3]. For Kirchhoff boundary conditions the elements of the scattering matrix are given by − , ,  = 1,..., n,see [3, Eq. (1)] (for the star-graph C = 1), and [4, Ex. 2.1.7, p. 41]. This is the approach used (for compact bb graphs) by Kottos and Smilansky in [17] and in several other works, see, e.g., [3], the review [14], and the monograph [4]. We have already noted that, up to a sign, the coefficients − coincide with the elements of the matrix S identifying both the classical and quantum scattering operators. Compared to this approach, we followed a different train of thought, starting from the trivial dynamics of a classical particle on the half-line with elastic collision at the origin and making use of a Kre˘ın-type resolvent formula (see [20,21]) to find a suitable singular perturbation of the Liouville operator associated with such a trivial dynamics. We remark that, in a similar way, one can reconstruct the Hamiltonian H starting from the Hamiltonian of a quantum particle on the half-line with Dirichlet boundary conditions at the origin. We defined the generator of the trivial dynamics on the half-line through Eq. (1.8). −it L Note that if f ∈ dom(L ), then f = e f satisfies the Liouville equation D t i f = L f , t D ∂t but the action of the group can be extended in a natural way to any bounded function. Since the evolution is unitary in L (R × R),if 2 = 1, we can interpret L (R ×R) −it L (q, p) := e f (q, p) as a probability density in the phase space R × R. Setting (q, p) := | f (q, p)| ,for all t ∈ R one has that pt pt ⎨ q − , p if q − > 0 m m (q, p) = (5.1) pt pt − q + , −p if q − < 0 , m m and it satisfies the equation ρ ρ i =−iX ·∇ , t 0 t ∂t pt for all f ∈ C (R × R) and q − = 0. m The semiclassical limit on a star-graph with Kirchhoff… Page 37 of 43 45 We remark that, assuming elastic collision at the origin, a classical particle moving on the half-line follows a simple, though discontinuous, trajectory in the phase space: at any time t ∈ R any initial state (q, p) ∈ R × R is mapped to pt pt ⎨ q + , p if q + > 0 , m m (q, p) := (5.2) pt pt − q − , −p if q + < 0 . m m Hence, given a density : R × R → R in the phase space, one has the identity + + −it L ρ ρ ϕ (q, p) = ( (q, p)) [see Eqs. (5.1) and (5.2)]. In this sense, the group e t −t given in Eq. (1.8) describes a classical particle on the half-line. The function f := −it L e f should be interpreted as a classical wave function, with associated probability density function in the phase space given by (q, p) := f (q, p) . t t −it L On a graph this interpretation fails when the generator of the dynamics is e . In particular, from Eq. (1.9), for all t ∈ R it follows ⎪ pt pt f q − , p if q − > 0 , m m −it L e f (q, p) = ⎪ pt pt (S) f − q + , −p if q − < 0 , ⎪ , m m −it L but the density (q, p) = e f (q, p) cannot be understood in terms of a ,t trajectory of a classical particle since in this case there is no phase space flow such ρ ρ ϕ that (q, p) = ( (q, p)). Also, such a density does not coincide with the time t −t evolution of the initial density | f | as prescribed by Barra and Gaspard, see [3,Eq. (10)]. The latter, adapted to our setting and notation, and taking into account the fact that we are considering a non-compact graph, would give (for all t ∈ R) pt pt f q − , p if q − > 0 , m m BG (q, p) = ,t pt pt ⎪ |(S) | f − q + , −p if q − < 0 . m m Additionally, we remark that the initial state  (ξ) does not define a probability σ,x density by the relation (ξ) =  (ξ) , because, in general, it does not belong σ,x to L (R × R). Nevertheless, the phase space evolution of the approximated coherent states  (ξ) σ,x induced by the Liouville operator L turns out to be a useful tool for the investigation of the semiclassical limit of the quantum evolution on the graph. 45 Page 38 of 43 C. Cacciapuoti et al. 5.2 Coherent states on a star-graph with an even number of edges Obviously, by superposition, one could consider initial states of the form ⎛ ⎞ σ ,ξ ⎜ ⎟ ⎜ ⎟ ≡ , σ σ ( ,ξ ),...,( ,ξ ) . 1 1 n n ⎝ ⎠ σ ,ξ n n with σ > 0 and ξ = (q , p ) ∈ R × R, and results similar to the ones stated in Theorems 1.4 and 1.6 hold true (with additive error terms). For n even it is possible to construct states that propagate exactly like coherent states on the real line. Suppose n even and consider a state ψ defined component-wisely by σ,ξ ψ if  = 1,..., n/2 , σ,ξ (ψ ) := σ ξ ψ if  = n/2 + 1,..., n . ,− ξ It is easy to check that such states belong to the domain of H . Next, consider the state e ψ . Taking the time derivative component by component one has σ ,ξ ∂ i   i A A t t ˚ ˚ i  e (ψ ) =− e (ψ ) , σ ,ξ σ ,ξ t t t t ∂t 2m t ˚ by the definition of coherent states, see Eq. (1.1). Since e ψ ∈ dom(H ) the σ ,ξ latter is equivalent to ∂ i i A A t t ˚ ˚ i  e ψ = H e ψ . σ ,ξ σ ,ξ t t t t ∂t i   t  i A −i H A t ˚ ˚ K ˚ t ˚ Moreover e ψ = ψ . Hence, e ψ = e ψ . On the other σ ,ξ σ ,ξ σ ,ξ σ ,ξ t t t =0 0 0 t t ˚ ˚ ˚ hand, define the classical state φ , component-wisely, by (φ ) (ξ) := (ψ ) (x ). σ,x σ,x σ,ξ ˚ ˚ Noting the identity − (φ ) (− ξ) = (φ ) (ξ), and by using Eq. δ, σ,x σ,x =1 n it L K ˚ ˚ ˚ (1.9), it is easy to infer the identity e φ (ξ) = φ (ξ ) = ψ , so that σ ,(·) σ ,(·) t σ ,ξ t t t t t i −i H A it L K ˚ K ˚ e ψ = e e φ (ξ), σ ,ξ σ ,(·) 0 t which is equivalent to (1.2). In this sense, up to a normalization factor 2/n, states of the form ψ are coherent states on a star-graph with an even number of edges. σ,ξ Funding Open Access funding provided by Università degli Studi dell’Insubria Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. The semiclassical limit on a star-graph with Kirchhoff… Page 39 of 43 45 Compliance with ethical standards Conflict of interest No potential competing interest was reported by the authors. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. A Appendix: Wave operators for the pair (Dirichlet Laplacian, Neumann Laplacian) on the half-line Proposition A.1 Let  be the wave operators for the pair (Dirichlet Laplacian, ND Neumann Laplacian) in L (R ), defined by ± N D := s-lim U U . t →±∞ −t t ND There holds true: ± ∗ =±F F . ND c Proof We use a density argument. Let ψ ∈ L (R ). For any ε > 0 there exists ∈ D/N C (R ) such that ψ − ≤ ε /4. Recalling the trivial bounds ≤ 1 L (R ) t 0 + and ≤ 1, we infer s/c N D ∗ N D ∗ (U U ∓ F F ) ψ 2 ≤ + (U U ∓ F F ) 2 . s s L (R ) L (R ) −t t c −t t c + + Hence, it is enough to prove that N D ∗ ∞ ϕ ϕ lim (U U ∓ F F ) = 0 ∀ ∈ C (R ). (A.1) s + −t t c L (R ) 0 t →±∞ Note that N D ∗ D N ∗ ϕ ϕ (U U ∓ F F ) (U ∓ U F F ) 2 2 s s −t t c L (R ) t t c L (R ) + + D ∗ N ∗ (U F ∓ U F )F 2 . t s t c L (R ) Moreover, for all t ∈ R the following identities hold true: 2i 2 D −ik t (U ψ)(x ) = √ dk e sin(kx)(F ψ)(k) ; 2 π 0 45 Page 40 of 43 C. Cacciapuoti et al. N −ik t (U ψ)(x ) = dk e cos(kx)(F ψ)(k). 2 π 0 Hence, 2i 2 D ∗ −ik t ψ ψ (U F )(x ) = √ dk e sin(kx ) (k), t s 2 π 2 2 N ∗ −ik t (U F ψ)(x ) = √ dk e cos(kx ) ψ(k), t c 2 0 which give 2 2 D ∗ N ∗ −ik t ϕ ϕ (U F ∓ U F )F (x ) = √ dk (i sin(kx )∓cos(kx )) e (F )(k) s s t s t c 2 0 2 2 ∓ikx −ik t =∓ √ dk e (F )(k). 2 π We have obtained the following explicit formula for the quantity we are interested in ∞ ∞ 2 2 N D ∗ 2 ∓ikx −ik t ϕ ϕ (U U ∓ F F ) = dx dk e (F )(k) . s s −t t c L (R ) 0 0 We note that, to prove the statement for  we have to study the limit t →+∞ ND of ∞ ∞ 2 2 N D ∗ 2 −ikx −ik t ϕ ϕ (U U − F F ) = dx dk e (F )(k) , (A.2) s s −t t c L (R ) 0 0 while, to prove the statement for  we have to study the limit t →−∞ of ND ∞ ∞ N D ∗ 2 ikx +ik |t | ϕ ϕ (U U + F F ) = dx dk e (F )(k) . s 2 s −t t c L (R ) 0 0 In what follows we focus the attention on the limit t →+∞. The other limit is obtained with trivial modifications. Hence, from now on we assume t > 0. From Eq. 1/2 1/2 (A.2), changing variables k → = kt and x → y = x /t , we obtain ∞ ∞ 2 2 N D ∗ 2 −i η y−i η 1/2 ϕ η ϕ η (U U − F F ) = dy d e (F )( /t ) s s −t t c 2 L (R ) 1/2 π t 0 0 1 ∞ 2 2 1 2 2 = dy |F (y, t )| + dy |F (y, t )| , 1/2 1/2 π t π t 0 1 with η η −i y−i 1/2 η ϕ η F (y, t ) := d e (F )( /t ). 0 The semiclassical limit on a star-graph with Kirchhoff… Page 41 of 43 45 ∞ ∞ ϕ ϕ For any ∈ C (R ), F belongs to C (R ), it decays at infinity faster than + s + any polynomial in 1/k, (F )(0) = 0, moreover 2k ϕ ϕ |(F )(k)|≤ √ dx x | (x )| . 2 π Hence, dk |F (k)| < ∞ for all < 2 . (A.3) ∞ ∞ ϕ ϕ Additionally, for any ∈ C (R ), F belongs to C (R ), it decays at infinity + c + ϕ √ ϕ faster than any polynomial in 1/k, and |(F )(k)|≤ dx | (x )|. Hence, 2 π dk |F (k)| < ∞ for all < 1 . (A.4) c δ 2 2 η η i d η η −i y−i −i y−i Starting from the identity e = e ,byintegrationbyparts, η η y+2 d we obtain F (y, t ) = F (y, t ) + F (y, t), 1 2 with 2 2i η η −i y−i 1/2 η η F (y, t ) := d e (F )( /t ) 1 s (y + 2 ) and 1 2 1 η η −i y−i 1/2 η ϕ η F (y, t ) := d e F ((·) ) ( /t ). 2 c 1/2 t i (y + 2 ) 1 1 1 Since y and are both positive, from the trivial inequality ≤ for a c η η (y+2 ) y (2 ) all y, > 0 and for all a, b, c > 0 such that a = b + c, we deduce that C 1 1/2 η ϕ η |F (y, t )|≤ d |(F )( /t )| 1 s 1/4 7/4 C 1 C ξ ξ = d |(F )( )|= ; 3/8 1/4 7/4 3/8 1/4 t y ξ t y here and in the following C denotes a generic positive constant whose value may ϕ ϕ depend only on integrals of the sine (or cosine) Fourier transform of (or (·) ), as in Eqs. (A.3) and (A.4); the value of C may change from line to line and precise values for the constants can be obtained. In a similar way we infer, C 1 1/2 η ϕ η |F (y, t )|≤ d F ((·) ) ( /t ) 2 c 1/2 1/4 3/4 t y 0 45 Page 42 of 43 C. Cacciapuoti et al. C 1 C = d ξ F ((·) ) (ξ) = . 3/8 1/4 3/4 3/8 1/4 t y ξ t y Hence, 1 1 2 C 1 C | | dy F (y, t ) ≤ dy ≤ . 1/2 5/4 1/2 5/4 t t y t 0 0 On the other hand, for y > 1, ∞ ∞ C 1 C 1 C 1/2 η ϕ η ϕ |F (y, t )|≤ d |(F )( /t )|= d ξ |(F )(ξ)|= 1 s s y y ξ y 0 0 and ∞ ∞ C C C 1/2 η ϕ η ϕ |F (y, t )|≤ d F ((·) ) ( /t ) = d ξ F ((·) ) (ξ) = . 2 c c 1/2 t y y y 0 0 So, we obtain ∞ ∞ 2 C 1 C dy |F (y, t )| ≤ dy ≤ . 1/2 1/2 2 1/2 t t y t 1 1 In this way we have proved that, for all ∈ C (R ) there exists a constant C such that N D ∗ (U U − F F ) 2 ≤ for all t > 1 , s L (R ) −t t c + 1/4 and the latter claim gives the limit in Eq. (A.1)for t →+∞. References 1. Adami, R., Cacciapuoti, C., Finco, D., Noja, D.: Fast solitons on star graphs. Rev. Math. Phys. 23(04), 409–451 (2011) 2. F. Barra, P. Gaspard: Transport and dynamics on open quantum graphs. Phys. Rev. E 65 (2001), 016205 (21 pages) 3. F. Barra, P. Gaspard: Classical dynamics on graphs. Phys. Rev. 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