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D. Damanik, G. Teschl
Institute for Mathematical Physics Bound States of Discrete Schrödinger Operators with Super–critical Inverse Square Potentials Bound States of Discrete Schr¨odinger Operators with Super-critical Inverse Square Potentials
We consider the family H := ΔΔ − μV,μ ∈ R, of discrete Schrödinger-type operators in d-dimensional lattice Z , where Δ is the discrete Laplacian and V is of rank-one. We prove that there exist coupling constant o o thresholds μ ,μ ≥ 0 such that for any μ ∈[−μ ,μ ] the discrete spectrum of H o o μ is empty and for any μ ∈ R \[−μ ,μ ] the discrete spectrum of H is a singleton o μ 2 o {e(μ)}, and e(μ) < 0for μ>μ and e(μ) > 4d for μ< −μ . Moreover, we study the asymptotics of e(μ) as μ μ and μ −μ as well as μ →±∞. The asymptotics highly depends on d and V . Keywords Discrete bilaplacian · Essential spectrum · Discrete spectrum · Eigenvalues · Asymptotics · Expansion Mathematics Subject Classification 47A10 · 47A55 · 47A75 · 41A60 Communicated by Gerald Teschl. Shokhrukh Yu. Kholmatov shokhrukh.kholmatov@univie.ac.at Ahmad Khalkhuzhaev ahmad_x@mail.ru Mardon Pardabaev p_mardon75@mail.ru University of Vienna, Oskar-Morgenstern Platz 1, Vienna 1090, Austria Samarkand State University, University boulevard 3, Samarkand, Uzbekistan 140104 123 608 S. Yu. Kholmatov et al. 1 Introduction In this paper we investigate the spectral properties of the perturbed discrete biharmonic operator H := ΔΔ − μV,μ ∈ R, (1.1) in the d-dimensional cubical lattice Z , where Δ is the discrete Laplacian and V is a is rank-one potential with a generating potential v. This model is associated to a one-particle system in Z with a potential field v, in which the particle freely “jumps” from a node X of the lattice not only to one of its nearest neighbors Y (similar to the discrete Laplacian case), but also to the nearest neighbors of the node Y . From the mathematical point of view, the discrete bilaplacian represents a discrete Schrödinger operator with a degenerate bottom, i.e., ΔΔ is unitarily equivalent to a multiplication operator by a function e which behaves as o(| p − p | ) close to its minimum point p . The spectral properties of discrete Schrödinger operators with non-degenerate bot- tom (i.e., e behaves as O(| p − p | ) close to its minimum point p ), in particular 0 0 with discrete Laplacian, have been extensively studied in recent years (see e.g. [1, 2, 7, 8, 10, 11, 20, 21, 23, 26, 28] and references therein) because of their applications in the theory of ultracold atoms in optical lattices [16, 24, 35, 36]. In particular, it is well-known that the existence of the discrete spectrum is strongly connected to the threshold phenomenon [18, 20–22], which plays an role in the existence the Efimov effect in three-body systems [31, 32, 34]: if any two-body subsystem in a three-body system has no bound state below its essential spectrum and at least two two-body subsystem has a zero-energy resonance, then the corresponding three-body system has infinitely many bound states whose energies accumulate at the lower edge of the three-body essential spectrum. Recall that the Efimov effect may appear only for certain attractive systems of particles [29]. However, recent experimental results in the theory of ultracold atoms in an optical lattice have shown that two-particle systems can have repulsive bound states and resonances (see e.g. [36]), thus, one expects the Efimov effect to hold also for some repulsive three-particle systems in Z . The strict mathematical justification of the Effect effect including the asymptotics for the number of negative eigenvalues of the three-body Hamiltonian has been suc- 3 3 cessfully established in 3-space dimensions (for both R and Z ) see e.g., [1, 4, 13, 19, 29, 31, 32, 34] and the references therein. In particular, the non-degeneracy of the bottom of the (reduced) one-particle Schrödinger operator played an important role in the study of resonance states of the associated two-body system [1, 31]. Another keypoint in the proof of the Efimov effect in Z was the asymptotics of the (unique) smallest eigenvalue of the (reduced) one-particle discrete Schrödinger operator which creates a singularity in the kernel of a Birman-Schwinger-type operator which used to obtain an asymptotics to the number of three-body bound states. To the best of our knowledge, there are no published results related to the Efi- mov effect in lattice three-body systems in which associated (reduced) one-body Schrödinger operator has degenerate bottom. 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 609 We also recall that fourth order elliptic operators in R in particular, the biharmonic operator, play also a central role in a wide class of physical models such as linear elasticity theory, rigidity problems (for instance, construction of suspension bridges) and in streamfunction formulation of Stokes flows (see e.g. [9, 25, 27] and references therein). Moreover, recent investigations have shown that the Laplace and biharmonic operators have high potential in image compression with the optimized and sufficiently sparse stored data [15]. The need for corresponding numerical simulations has led to a vast literature devoted to a variety of discrete approximations to the solutions of fourth order equations [5, 12, 33]. The question of stability of such models is basically related to their spectral properties and therefore, numerous studies have been dedicated to the numerical evaluation of the eigenvalues [3, 6, 30]. The aim of the present paper is the study of the existence and asymptotics of eigenvalues as well as threshold resonance and bound states of H defined in (1.1), which corresponds to the one-body Schrödinger operator with degenerate bottom. Namely, we study the discrete spectrum of H depending on μ and on v. For simplicity we assume the generator v of V to decay exponentially at infinity, however, we urge that our methods can also be adjusted to less regular cases (see Remark 2.6). Since the spectrum of Δ consists of [0, 2d] (see e.g., [1]), by the compactness of V and Weyl’s Theorem, the essential spectrum of H fills the segment [0, 4d ] independently of μ. Moreover, the essential spectrum does not give birth to a new eigenvalue while μ runs in some real interval [−μ ,μ ], and it turns out as soon as μ leaves this interval through μ resp. through −μ , a unique negative resp. a unique positive eigenvalue e(μ) releases from the essential spectrum (Theorem 2.2). Now we are interested in the absorption rate of e(μ) as μ → μ and μ →−μ . The associated asymptotics are highly dependent not only on the dimension d of the lattice (as in the discrete Laplacian case [20, 21]), but also values on the multiplicity 2n and 2n of 0 ∈{v = 0} (if v(0) = 0) and π ∈{v = 0} (if v(π) = 0), respectively. More precisely, depending on d and n , e(μ) has a convergent expansion 1/3 –in (μ − μ ) for 2n + d = 1, 7; o o –in μ − μ for 2n + d = 3, 5; o o 1/4 –in (μ − μ ) for 2n + d ≥ 9 with d odd; o o –in μ − μ and −(μ − μ ) ln(μ − μ ) for 2n + d = 2, 6; o o o o −1/(μ−μ ) –in μ − μ and e for 2n + d = 4; o o −1 1 ln ln(μ−μ ) 1/2 1/2 o –in (μ − μ ) , −(μ − μ ) ln(μ − μ ), (− ) and − for o o o ln(μ−μ ) ln(μ−μ ) o o 2n + d = 8; 1/2 1/2 o –in (μ − μ ) and −(μ − μ ) ln(μ − μ ) for 2n + d ≥ 10 with d even o o o (see Theorem 2.4). Moreover, resonance states of 0-energy, i.e. non-zero solutions f 2 d of H f = 0 not belonging to (Z ) appear if and only if 2n + d ∈{5, 6, 7, 8}. μ o Recall that the emergence of 0-energy resonances in more lattice dimensions could allow the Efimov effect to be observed in other dimensions than d = 3. Furthermore, observing that the top e(π) = 4d of the essential spectrum is non- degenerate, one expects the asymptotics of e(μ) as μ →−μ to be similar as in the discrete Laplacian case [20, 21]; more precisely, depending on d and n , e(μ) has a convergent expansion o o –in μ + μ for 2n + d = 1, 3; 123 610 S. Yu. Kholmatov et al. o 1/2 o –in (μ + μ ) for 2n + d ≥ 5 with d odd; o −1/(μ+μ ) –in μ + μ and e for 2n + d = 2; o −1 1 ln ln(μ+μ ) –in μ + μ , − and − for 2n + d = 4; o o o ln(μ+μ ) ln(μ+μ ) o o o o –in μ + μ and −(μ + μ ) ln(μ + μ ) for 2n + d ≥ 6 with d even (see Theorem 2.5). Moreover, the resonance states of energy 4d , i.e. non-zero solu- 2 2 d tions f of H f = 4d f not belonging to (Z ) appear if and only if 2n +d = 3, 4. −μ o The threshold analysis for more general class of nonlocal discrete Schrödinger operators with δ-potential of type H = (−Δ) + μδ , μ x0 can be found in [14], where is some strictly increasing C -function and δ is the x0 Dirac’s delta-function supported at 0. Besides the existence of eigenvalues, authors of [14] classify (embedded) threshold resonances and threshold eigenvalues depending on the behaviour of at the edges of the essential spectrum of −Δ and on the lattice dimension d. The eigenvalue expansions for the discrete bilaplacian with δ- perturbation have been established in [17]for d = 1 using the complex analytic methods. The paper is organized as follows. In Sect. 2 after introducing some preliminaries we state the main results of the paper. In Theorem 2.2 we establish necessary and sufficient conditions for non-emptiness of the discrete spectrum of H , and in case of existence, we study the location and the uniqueness, analiticity, monotonicity and convexity properties of eigenvalues e(μ) as a function of μ. In particular, we study the asymptotics of e(μ) as μ → μ and μ →−μ as well as μ →±∞. As discussed above in Theorems 2.4 and 2.5 we obtain expansions of e(μ) for small and positive μ − μ and μ + μ . In Sect. 3 we prove the main results. The main idea of the proof is to obtain a nonlinear equation (μ; z) = 0 with respect to the eigenvalue z = e(μ) of H and then study properties of (μ; z). Finally, in appendix Section A we obtain the asymptotics of certain integrals related to (μ; z) which will be used in the proofs of main results. Data availability statement We confirm that the current manuscript has no associated data. 2 Preliminary and main results d 2 d Let Z be the d-dimensional lattice and (Z ) be the Hilbert space of square- summable functions on Z . Consider the family H := H − μV,μ ≥ 0, μ 0 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 611 2 d of self-adjoint bounded discrete Schrödinger operators in (Z ). Here H := ΔΔ is discrete bilaplacian, where 2 d Δ f (x ) = [ f (x ) − f (x + s)], f ∈ (Z ), |s|=1 is the discrete Laplacian, and V is a rank-one operator V f (x ) = v(x ) v(y) f (y), y∈Z 2 d where v ∈ (Z ) \{0} is a given real-valued function. d 2 d Let T be the d-dimensional torus equipped with the Haar measure and L (T ) be the Hilbert space of square-integrable functions on T . By F we denote the the standard Fourier transform 2 d 2 d ixp F : (Z ) → L (T ), F f (p) = f (x )e . d/2 (2π) x ∈Z Further we always assume that v and its Fourier image ix · p v(p) := F v(p) = v(x )e d/2 (2π) x ∈Z satisfy the following assumptions: There exist reals C , a > 0 and nonnegative integersn , n ≥ 0 such that −a|x | d | v(x )|≤ Ce for all x ∈ Z , (H1) 2 2 2 2n −2 2 2n 2 o o |v(0)| = D |v(0)| = ... = D |v(0)| = 0, D |v(0)| = 0, (H2) o o 2 2 2 2n −2 2 2n 2 |v(π) | = D |v(π) | = ... = D |v(π) | = 0, D |v(π) | = 0, (H3) here D f (p) is the j-th order differential of f at p, i.e. the j-th order symmetric tensor ∂ f (p) i i j 1 d D f (p)[w,...,w]= w ...w , 1 d i i 1 d ∂ p ...∂ p 1 d i +...+i = j ,i ≥0 1 d k j −times w = (w ,...,w ) ∈ R , 1 d d d and π = (π,...,π) ∈ T . Notice that under assumption (H1), v is analytic on T . Recall that σ(Δ) = σ (Δ) =[0, 2d] (see e.g. [1]). Hence, σ(H ) = σ (H ) = ess 0 ess 0 [0, 4d ], and by the compactness of V and Weyl’s Theorem, σ (H ) = σ (H ) =[0, 4d ] ess μ ess 0 123 612 S. Yu. Kholmatov et al. for any μ ∈ R. Before stating the main results let us introduce the constants 2 2 −1 −1 |v(q)| dq |v(q)| dq μ := ,μ := , (2.2) d d e(q) 4d − e(q) T T 2 2 |v(q)| dq |v(q)| dq c := , C := , (2.3) v v 2 2 2 d e(q) d (4d − e(q)) T T and 2n +d 2n 2 d−1 c := D |v(0)| [w, . . . , w] dH (w), (2.4) d−1 (2n )! 2n +d−1 2 o 2n 2 d−1 C := D |v(π) | [w, . . . , w] dH (w), (2.5) n +d/2 o d−1 (8d) (2n )! d−1 d where S is the unit sphere in R and e(q) := (1 − cos q ) . i =1 Remark 2.1 Under assumptions (H1)–(H3), μ ,μ ≥ 0, c , C > 0, and c , C ∈ o v v v v (0, +∞]. Moreover, by Propositions A.1 and A.2: o o – μ = 0 (resp. μ = 0) if and only if 2n + d ≤ 4 (resp. 2n + d ≤ 2); o o – c < ∞ (resp. C < ∞)if2n + d ≥ 9 (resp. 2n + d ≥ 5). v v o 2.1 Main results First we concern with the existence of the discrete spectrum of H . Theorem 2.2 Let μ ,μ ≥ 0 be given by (2.2). Then σ (H ) =∅ for any μ ∈ o disc μ o o [−μ ,μ ] and σ (H ) is a singleton {e(μ)} for any μ ∈ R \[−μ ,μ ]. Moreover, o disc μ o the associated eigenfunction f to e(μ) is given by f := F f , where μ μ μ v(p) f (p) = . e(p) − e(μ) o 2 Furthermore, if μ< −μ (resp. μ>μ ), then e(μ) > 4d (resp. e(μ) < 0). Moreover, the function μ ∈ R\[−μ ,μ ] → e(μ) is real-analytic strictly decreasing, convex in (−∞, −μ ) and concave in (μ , +∞), and satisfies lim e(μ) = 0 and lim e(μ) = 4d (2.6) μμ μ−μ 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 613 and e(μ) lim =− |v(q)| dq. (2.7) μ→±∞ μ d Next we study the threshold resonances of H . Theorem 2.3 Let n , n ≥ 0 be given by (H2)–(H3). ∗ d (a) Let 2n + d ≥ 5. Then f := F f ∈ c (Z ), i.e., f (x ) → 0 as |x|→+∞, o 0 where v(p) 1 d f (p) = ∈ L (T ). e(p) d 2 d 2 d Moreover, f ∈ c (Z ) \ (Z ) for 2n + d ∈{5, 6, 7, 8}, f ∈ (Z ) for 0 o 2n + d ≥ 9, and f solves the equation H f = 0. o μ o ∗ 0 d (b) Let 2n + d ≥ 3. Then g := F g ∈ (Z ), where v(p) g(p) = . 4d − e(p) 0 d 2 d o 2 d o Moreover, g ∈ (Z ) \ (Z ) for 2n + d ∈{3, 4}, g ∈ (Z ) for 2n + d ≥ 5, and g solves the equation H f = 4d f . −μ We recall that in the literature the non-zero solutions of equations H f = 0 and 2 2 d H g = 4d g not belonging to (Z ) are called the resonance states [1, 2]. Now we study the rate of the convergences in (2.6). Theorem 2.4 (Expansions of e(μ) at μ = μ ) For μ>μ let e(μ) < 0 be the o o eigenvalue of H . (a) Suppose that d is odd: (a1) if 2n + d = 1, 3, then μ = 0 and for sufficiently small and positive μ, o o 1/3 n+1 πc 1/3 μ + c μ , 2n + d = 1, 1,n o 1/4 n≥1 (−e(μ)) = πc v n+1 ⎩ μ + c μ , 2n + d = 3, 3,n o n≥1 where {c } and {c } are some real coefficients; 1,n 3,n (a2) if 2n + d = 5, 7, then μ > 0 and for sufficiently small and positive μ − μ , o o o 1/4 (−e(μ)) 8 n+1 (μ − μ ) + c (μ − μ ) , 2n + d = 5, ⎪ o 5,n o o ⎨ πc μ n≥1 = 1/3 n+1 8 1/3 ⎩ (μ − μ ) + c (μ − μ ) , 2n + d = 7, o 7,n o o πc μ n≥1 where {c } and {c } are some real coefficients; 5,n 7,n 123 614 S. Yu. Kholmatov et al. (a3) if 2n + d ≥ 9, then μ > 0 and for sufficiently small and positive μ − μ , o o o 1/4 2 −1/4 1/4 n/4 (−e(μ)) = (μ c ) (μ − μ ) + c (μ − μ ) , v o 9,n o n≥1 where {c } are some real coefficients. 9,n (b) Suppose that d is even: (b1) if 2n + d = 2, 4, then μ = 0 and for sufficiently small and positive μ, o o 1/2 (−e(μ)) πc n+1 m μ + c μ (−μ ln μ) , 2n + d = 2, ⎪ 2,nm o n+m≥1,n,m≥0 = m+1 8 8 − − ⎪ n+1 c μ c μ v v ⎪ ce + c μ e , 2n + d = 4, 4,nm o n+m≥1,n,m≥0 where {c } and {c } are some real coefficients and c > 0; 2,nm 4,nm (b2) if 2n + d = 6, 8, then μ > 0 and for sufficiently small and positive μ − μ , o o o 1/2 (−e(μ)) 2 2n+2 m τ + c τ θ , 2n + d = 6, ⎪ 6,nm o ⎨ πc μ n+m≥1,n,m≥0 1/2 n+1 m+1 k τσ + c τ σ η , 2n + d = 8, 8,nmk o c μ n+m+k≥1,n,m,k≥0 where {c } and {c } are some real coefficients and 4,nm 8,nmk −1 1/2 1 ln ln τ 1/2 2 τ := (μ − μ ) ,θ := −τ ln τ, σ := − ,η := − , ln τ ln τ (2.8) (b3) if 2n + d ≥ 10, then μ > 0 and for sufficiently small and positive μ − μ , o o o 1/2 2 −1/2 n+1 m (−e(μ)) = (μ c ) τ + c τ θ , v 10,nm n+m≥1,n,m≥0 where {c } are some real coefficients. 10,nm Here c > 0 and c > 0 are given by (2.4) and (2.3), respectively. v v o o 2 Theorem 2.5 (Expansions of e(μ) at μ =−μ )For let μ< −μ let e(μ) > 4d be the eigenvalue of H . (a) Suppose that d is odd: 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 615 o o (a1) if 2n + d = 1, then μ = 0 and for sufficiently small and negative μ, 2 1/2 n+1 (e(μ) − 4d ) =−πC μ + C μ , v 1,n n≥1 where {C } are some real coefficients; 1,n o o o (a2) if 2n + d = 3, then μ > 0 and for sufficiently small and positive μ + μ , 2 1/2 o2 −1 o o n+1 (e(μ) − 4d ) = (πC μ ) (μ + μ ) + C (μ + μ ) , v 3,n n≥1 where {C } and {C } are some real coefficients; 3,n 7,n o o o (a3) if 2n + d ≥ 5, then μ > 0 and for sufficiently small and positive μ + μ , 2 1/2 o −1/2 o 1/2 o (n+1)/2 (e(μ) − 4d ) = (C μ ) (μ + μ ) + C (μ + μ ) , v 5,n n≥1 where {C } are some real coefficients. 5,n (b) Suppose that d is even: (b1) if 2n + d = 2, then μ = 0 and for sufficiently small and negative μ, o o m+1 1 1 2 n+1 C μ C μ v v e(μ) − 4d = Ce + C μ − e , 2,nm n+m≥1,n,m≥0 where {C } are some real coefficients and C > 0; 2,nm o o (b2) if 2n + d = 4, then μ > 0 and for sufficiently small and positive μ + μ , 2 o −1 n+1 m+1 k e(μ) − 4d = (C μ ) μσ + C τ σ η , v 4,nmk n+m+k≥1,n,m,k≥0 where {C } are some real coefficients and 4,nm −1 1 ln ln τ τ := μ + μ,σ := − ,η := − ; ln τ ln τ o o (b3) if 2n + d ≥ 6, then μ > 0 and for sufficiently small and positive μ + μ , 2 o −1 o e(μ) − 4d = (C μ ) (μ + μ ) o n+1 o o m + C (μ + μ ) [−(μ + μ ) ln(μ + μ )] , 6,nm n+m≥1,n,m≥0 where {C } are some real coefficients. 6,nm Here C and C are given by (2.5) and (2.3), respectively. v v 123 616 S. Yu. Kholmatov et al. Remark 2.6 Few comments on the main results are in order. 2 d 1. The assertions of Theorem 2.2 hold in fact for any v ∈ (Z ) (see Remark 3.2); 2. Similar expansions of e(μ) in Theorems 2.4 and 2.5 at μ = μ and μ =−μ , respectively, still hold for any exponentially decaying v : Z → C (see Remark 3.3); −α 3. If v decays at most polynomially at infinity, i.e. v(x ) = O(|x | ) for some α> 0, then instead of the expansions in Theorem 2.4 and 2.5 we obtain only asymptotics of e(μ) (see Remark 3.4). 3 Proof of main results In this section we prove the main results. By the Birman-Schwinger principle and the Fredholm Theorem we have Lemma 3.1 A complex number z ∈ C \[0, 4d ] is an eigenvalue of H if and only if |v(q)| dq (μ; z) := 1 − μ = 0. d e(q) − z Proof of Theorem 2.2 By the definition of μ , for any μ< −μ : lim (μ; z) = 1 + < 0, lim (μ; z) = 1. o o z−μ z→+∞ Since (μ; z)> 1for z < 0 and μ> −μ , in view of the strict monotonicity 2 o 2 (μ;·) in (4d , ∞), for any μ< −μ there exists a unique e(μ) ∈ (4d , +∞) such that (μ; e(μ)) = 0. Analogously, for any μ>μ there exists a unique e(μ) ∈ (−∞, 0) such that (μ; e(μ)) = 0. By the Implicit Function Theorem the function μ ∈ R \[−μ ,μ ] → e(μ) is real-analytic. Moreover, computing the derivatives of the implicit function e(μ) we find: 2 2 −1 1 |v(q)| dq |v(q)| dq e (μ) =− ,μ = 0, (3.1) d d μ e(q) − e(μ) (e(q) − e(μ)) T T thus, using μ(e(q) − e(μ)) > 0 we get e (μ) < 0, i.e. e(·) is strictly decreasing in R \{0}. Differentiating (3.1) one more time we get −1 2 2 2e (μ) |v(q)| dq |v(q)| dq e (μ) = 1 − μe (μ) . 3 2 μ d (e(q) − e(μ)) d (e(q) − e(μ)) T T Therefore, e (μ) > 0 (i.e. e(·) is strictly convex) for μ< 0 and e (μ) < 0 (i.e. e(·) is strictly concave) for μ> 0. To prove (2.7), first we let μ →±∞ in |v(q)| dq 1 = μ (3.2) d e(q) − e(μ) 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 617 e(q) 1 and find lim e(μ) =∓∞. In particular, if |μ| is sufficiently large, | | < and e(μ) 2 μ→±∞ hence, by (3.2) and the Dominated Convergence Theorem, e(μ) |v(q)| dq lim =− lim =− |v(q)| dq. e(q) μ→±∞ μ→±∞ d d T 1 − T e(μ) To prove that f solves H f = e(μ) f we consider the equivalent equality μ μ μ μ FH F f = e(μ) f , which is easily reduced to the equality (μ; e(μ)) = 0. μ μ μ Remark 3.2 In view of Lemma 3.1 and the proof of Theorem 2.2, their assertions still 2 d hold for any v ∈ (Z ). Proof of Theorem 2.3 We prove only (a), the proof of (b) being similar. Repeating the proof of the continuity (resp. differentiability) of l at z = 0 in Proposition A.1 one 1 d 2 d 2 d can show that f ∈ L (T ) \ L (T ) for 2n + d ∈{5, 6, 7, 8} and f ∈ L (T ) for 0 d 2n + d ≥ 9. Thus, by the Riemann-Lebesgue Lemma, f ∈ (Z ). To show that H f = 0 it suffices to observe that FH F f = 0. μ μ o 0 Proof of Theorem 2.4 Since ⎛ ⎞ ⎛ ⎞ 2 2 2 −d −d ⎝ ⎠ ⎝ ⎠ |v(p)| = (2π) v(x ) cos p · x + (2π) v(x ) sin p · x , d d x ∈Z x ∈Z (3.3) d 2 the function p ∈ T →|v(p)| is nonnegative even real-analytic function. Notice also that if n ≥ 1, then by the nonnegativity of |v| , p = 0 is a global minimum for 2 2n 2 |v| . Therefore, the tensor D |v(0)| is positively definite and 2n +d 2n 2 d−1 c := D |v(0)| [w,...,w]dH > 0. (2n )! d−1 o S Note that |v(q)| dq c = l (0) = , |v| 2 d e(q) v( p) 2 d where l is defined in (A.1). By Proposition A.1, f (p) = ∈ L (T ) if and only e( p) if 2n + d ≥ 9. Moreover, by definition, μ > 0 and (μ ; 0) = 0for 2n + d ≥ 5, o o o o and hence, as in the proof of Lemma 3.1 for such d one can show that H f = 0. In view of the strict monotonicity and (2.6) there exists a unique μ > 0 such that e(μ) ∈ (− , 0) for any μ ∈ (0,μ ). Since −1 μ = (l (e(μ))) , (3.4) |v| 123 618 S. Yu. Kholmatov et al. we can use Proposition A.1 with f =|v| and e := e(μ), to find the expansions of the inverse function μ := μ(e). Then applying the appropriate versions of the Implicit Function Theorem in analytical case we get the expansions of e = e(μ). Notice that from (A.3) and (A.4)aswellas(3.5) it follows that μ = 0for 2n + d ≤ 4 and o o −1 |v(q)| dq μ = > 0for 2n + d ≥ 5. o d o T e(q) (a) Suppose that d is odd. In view of the expansions (A.3)of l , in this case, (3.4) is reduced to the inverting the equation μ = g(α), (3.5) 1/4 where α := (−e) and g is an analytic function around α = 0. Case 2n + d = 1. In this case by (A.3), g(α) := , c + a α 1 n≥1 1/3 where {a }⊂ R and c := (πc /4) and (3.5) is equivalently represented as n 1 v ⎛ ⎞ 1/3 3 n ⎝ ⎠ α = μ c + a α , (3.6) n≥1 1/3 where μ = μ . Now setting α = μ(c + u), (3.7) 3 1/3 and using the Taylor series of (c + x ) , for μ and u sufficiently small we rewrite (3.6)as n n F (u,μ) := u − a ˜ μ (c + u) = 0, (3.8) n 1 n≥1 where F (·, ·) is analytic at (u,μ) = (0, 0), F (0, 0) = 0 and F (0, 0) = 1. Hence, by the Implicit Function Theorem, there exists γ > 0 such that for |μ| <γ , (3.8) has 1 1 a unique real-analytic solution u = u(μ) which can be represented as an absolutely convergent series u = b μ . Putting this in (3.7) and recalling the definitions n≥1 1/4 of α and μ we get the expansion of (−e(μ)) for μ> 0 small. Case 2n + d = 3. By (A.3), ⎛ ⎞ −1 ⎝ ⎠ g(α) = α c + a α , (3.9) 3 n n≥1 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 619 where {a }⊂ R and c := πc /8, and hence, (3.5) is represented as n 3 v ⎛ ⎞ ⎝ ⎠ α = μ c + a α . 3 n n≥1 Then setting α = μ(c + u) we rewrite (3.9)inthe form (3.8), and as in the case of 1/4 2n + d = 1, we get the expansion of (−e(μ)) . Case 2n + d = 5. In this case by (A.3) ⎛ ⎛ ⎞ ⎞ −1 1 πc α ⎝ ⎝ ⎠ ⎠ g(α) = − 1 + a α , μ 8 n≥1 where {a }⊂ R, and hence, by (3.5), ⎛ ⎞ μ − μ πc α o v ⎝ ⎠ = 1 + a α . (3.10) μμ 8 n≥1 Note that if |μ − μ | <μ , then o o μ − μ μ − μ μ − μ μ − μ o o o o = = , (3.11) 2 2 μμ μ + μ (μ − μ ) μ μ o o o o o o n≥0 thus from (3.10) we get ⎛ ⎞ ⎛ ⎞ −1 −n n n ⎝ ⎠ ⎝ ⎠ α = (μ − μ ) c + c μ (μ − μ ) 1 + a α . o 5 5 o n n≥1 n≥1 and c := 8/(πc μ ). Now setting α = (μ − μ )(c + u) for sufficiently small and 5 v o 5 positive μ − μ we get n m u = c˜ (μ − μ ) (c + u) , n,m o n,m≥1 where c˜ ⊂ R. By the Implicit Function Theorem, for sufficiently small μ−μ there n,m o exists a unique real-analytic function u = u(μ) given by the absolutely convergent series u(μ) = b (μ − μ ) . By the definition of α, this implies the expansion of n o n≥1 1/4 (−e(μ)) . 123 620 S. Yu. Kholmatov et al. Case 2n + d = 7. As the previous case, by (A.3) and (3.11), the equation (3.5)is represented as ⎛ ⎞ ⎛ ⎞ 3 3 −n n 3 n ⎝ ⎠ ⎝ ⎠ (μ − μ ) c + c μ (μ − μ ) = α 1 + a α , (3.12) o o n 7 7 o n≥1 n≥1 2 1/3 where {a }⊂ R and c := [8/(πc μ )] . When μ − μ > 0 is small enough, by n 7 v o ±1/3 the Taylor series of (1 + x ) at x = 0, (3.12) is equivalently rewritten as ⎛ ⎞ ⎛ ⎞ 1/3 n n ⎝ ⎠ ⎝ ⎠ α = (μ − μ ) c + c˜ (μ − μ ) 1 + a ˜ α , (3.13) o 7 n o n n≥1 n≥1 1/3 Thus, for ρ = (μ − μ ) , setting α = ρ(c + u) in (3.13), for sufficiently small o 7 and positive ρ we get n m u = c˜ ρ (c + u) . n,m 7 n,m≥1 By the Implicit Function Theorem, this equation has a unique real-analytic solution u = u(ρ) given by the absolutely convergent series u = b ρ . This, definitions n≥1 1/4 of α and ρ imply the expansion of (−e(μ)) . Case 2n + d = 9. In this case by (A.3) and (3.11) ⎛ ⎞ ⎛ ⎞ 4 4 −n n 4 n ⎝ ⎠ ⎝ ⎠ (μ − μ ) c + c μ (μ − μ ) = α 1 + a α , (3.14) o o n 9 9 o n≥1 n≥1 2 −1/4 where {a }⊂ R and c := (μ c ) . Thus, for sufficiently small and positive μ−μ n 9 v o ±1/4 using the Taylor series of (1 + x ) at x = 0, this equation can also be represented as ⎛ ⎞ ⎛ ⎞ 4n n ⎝ ˜ ⎠ ⎝ ⎠ α = ρ c + b ρ 1 + a ˜ α , 9 n n n≥1 n≥1 1/4 where ρ := (μ − μ ) . Now setting α = ρ(c + u) in (3.14) we get o 9 n m u = c˜ ρ (c + u) , n,m 9 n,m≥1 1/4 and the expansion of (−e(μ)) follows as in the case of 2n + d = 7. 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 621 (b) Suppose that d is even. In view of the expansion (A.3)of l , in this case, (3.4) is reduced to the inverting the equation μ = , (3.15) g(α) + h(α) ln α 1/2 where α := (−e) , l ∈ N , and g and h are analytic around α = 0. Presence of ln α implies that unlike the case of odd dimensions, α is not necessarily analytic with respect to μ . Therefore, we need to introduce new variables dependent on ln μ to reduce the problem to the Implicit Function Theorem. Case 2n + d = 2. By (A.4), in this case for c := πc /8 o 2 v n 2n l = 1, g(α) = c + a α , h(α) = b α . 2 n n n≥1 n≥1 Hence, setting α = μ(c + u) (3.16) and τ =−μ ln μ we represent (3.15)as n n n n n n F (u,μ,τ) := u − a μ (c + u) + ln(c + u) b μ (c + u) 2 2 2 n≥1 n≥1 n n−1 n − τ b μ (c + u) = 0, n≥1 where F is analytic around (0, 0, 0), F (0, 0, 0) = 0, F (0, 0, 0) = 1. Hence, by the Implicit Function Theorem, there exists a unique real-analytic function u = u(μ, τ ) n m given by the convergent series u(μ, τ ) = c˜ μ τ for sufficiently n,m n+m≥1,n,m≥0 small |μ| and |τ |, which satisfies F (u(μ, τ ), μ, τ ) ≡ 0. Inserting u in (3.16) we get 1/2 the expansion of α = (−e) . Case 2n + d = 4. In this case, by (A.4)for c := 8/c o 4 v 2n l = 0, g(α) = a α , h(α) =−c + b α . n n 4 n n≥0 n≥1 a /c c μ 0 4 Letting α = e (c + u), where c = e > 0, we represent (3.15)as 1 n−1 − − n n c μ c μ 4 4 ln(c + u) − b = e a e (c + u) n≥1 n n − − n n n n c μ c μ 4 4 + ln(c + u) b e (c + u) − a e (c + u) = 0. n≥1 n≥1 (3.17) 123 622 S. Yu. Kholmatov et al. 1 1 − − c μ c μ 4 4 Writing τ := e so that e = μτ, (3.17) is represented as n n−1 n−1 n F (u,μ,τ) := ln(c + u) − b − μ a μ τ (c + u) n≥1 n n n n n n n n − ln(c + u) b μ τ (c + u) + a μ τ (c + u) = 0, n≥1 n≥1 where F is analytic around (0, 0, 0), F (0, 0, 0) = 0, and F (0, 0, 0) = > 0. Thus, by the Implicit Function Theorem, for |μ|, |τ | and |u| small there exists a unique real analytic function u = u(μ, τ ) given by the convergent series u = n m c μ c˜ μ τ such that F (u(μ, τ ), μ, τ ) ≡ 0. Since τ = e , this n,m n+m≥1,n,m≥0 implies m+1 1 1 1 − − − n+1 c μ c μ c μ 4 4 4 α = e (c + u) = ce + c˜ μ e . n,m n+m≥1,n,m≥0 Case 2n + d = 6. In this case, by (A.4), for c := 8/(πc μ ) o 6 v ⎛ ⎞ 1 1 1 n 2n ⎝ ⎠ l = 0, g(α) = − α + a α , h(α) = b α , n n 2 2 μ c μ c μ o 6 6 o o n≥2 n≥1 and hence, (3.15) is represented as ⎛ ⎞ 1 1 1 n 2n ⎝ ⎠ − = α + a α + ln α b α , n n μ μ c μ o 6 n≥2 n≥1 or equivalently, by (3.11), μ − μ n 2n α = c (μ − μ ) − a α − ln α b α . (3.18) 6 o n n n≥0 n≥2 n≥1 Recalling the definitions of τ and θ in (2.8), setting α = τ (c + u), we represent (3.18)as 2n 2n−2 n F (u,τ,θ) := u − c − a τ (c + u) 6 n 6 n≥1 n≥2 4n 2n 4n−4 2n − ln(c + u) b τ (c + u) − θ b τ (c + u) = 0, 6 n 6 n 6 n≥1 n≥1 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 623 where F is real-analytic around (0, 0, 0), F (0, 0, 0) = 0 and F (0, 0, 0) = 1, and F is even in τ. Thus, by the Implicit Function Theorem, for |u|, |τ | and |θ | small there exists a unique real analytic function u = u(τ, θ ), even in τ, given by the convergent 2n m series u = c˜ τ θ such that F (u(τ, θ ), τ, θ ) ≡ 0. Thus, n,m n+m≥1,n,m≥0 2 2n+2 m α = τ (c + u) = c σ + c˜ τ θ . 6 6 n,m n+m≥1,n,m≥0 2 −1/2 Case 2n + d = 8. By (A.4), for c := [8/c μ ] , o 8 v ⎛ ⎞ 1 1 n 2 2n ⎝ ⎠ l = 0, g(α) = a α , h(α) = α + b α , n n 2 2 2 2 μ c μ c o 8 o 8 n≥2 n≥2 thus, as in the case of 2n + d = 6, (3.15) is represented as μ − μ 2 2 2n n c (μ − μ ) = α ln α + ln α b α + a α . (3.19) o n n n≥0 n≥2 n≥2 For τ, σ and η given in (2.8)set α = τσ (c + u) and represent (3.19)as 2n 2 2 n−1 n+1 n+2 2c u + u =c + a τ σ (c + u) 8 n 8 n≥1 n≥2 2n−2 2n+2 − b (τ σ ) (c + u) n 8 n≥2 ⎛ ⎞ 2 2 2n−2 2n+2 ⎝ ⎠ + σ ln(c + u) − (c + u) + b (τ σ ) (c + u) . 8 8 n 8 n≥2 This equation is represented as F (u,τ,σ,η) = 0, where F is real-analytic in a neighborhood of (0, 0, 0, 0), F (0, 0, 0, 0) = 0 and F (0, 0, 0, 0) = 2c > 0. u 8 Hence, for |u|, |τ |, |σ | and |η| small, by the Implicit Function Theorem, there exists a unique real-analytic function u = u(τ,σ,η) given by the convergent series n m k u = c˜ τ σ μ such that F (u(τ,σ,η), τ, σ, η) ≡ 0. Thus, n,m,k n+m+k≥1,n,m,k≥0 n+1 m+1 k α = τσ (c + u) = c τσ + c˜ τ σ η . 8 8 n,m,k n+m+k≥1,n,m,k≥0 2 −1/2 Case 2n + d ≥ 10. By (A.4)for c := (μ c ) , o 10 v 2 n+2 2n l = 0, g(α) = + c α + a α , h(α) = b α , v n n n≥2 n≥2 123 624 S. Yu. Kholmatov et al. and as in the case of 2n + d = 6, (3.15) is represented as μ − μ μ − μ o o 2 n+2 2n = c α + a α + ln α b α . (3.20) v n n μ μ n≥0 n≥2 n≥2 Recalling the definitions of τ and θ in (2.8), we set α = τ(c + u). Then (3.20)is represented as 2n 2 2 n n+2 F (u,τ,θ) :=2c u + u − c + a τ (c + u) 10 n 10 n≥1 n≥2 2n−4 2n 2n−2 2n − θ b τ (c + u) + ln(c + u) b τ (c + u) = 0, n 8 10 n 8 n≥2 n≥2 where F is analytic at (0, 0, 0), F (0, 0, 0) = 0 and F (0, 0, 0) = 2c > 0. u 10 Thus, by the Implicit Function Theorem, for |u|, |τ | and |θ | small there exists a unique real-analytic function u = u(τ, θ ) given by the convergent series u = n n c˜ τ θ such that F (u(τ, θ ), τ, θ ) ≡ 0. Then n,m n+m≥1,n,m≥0 n+1 n α = μ(c + u) = c μ + c˜ μ θ . 10 10 n,m n+m≥1,n,m≥0 Theorem is proved. d 2 Proof of Theorem 2.5 From (3.3) it follows that the map p ∈ T →|v| (π + p) is even. Now the expansions of e(μ) at μ =−μ can be proven along the same lines of Theorem 2.4 using Proposition A.2 with f =|v| . Remark 3.3 Let v : Z → C satisfy (H1). Since e(·) is even, |v(p)| dp 1 f (p)dp = , d d e(p) − z (2π) e(p) − z T T where ⎛ ⎞ ⎛ ⎞ 2 2 ⎝ ⎠ ⎝ ⎠ f (p) := v (x ) cos p · x + v (x ) cos p · x 1 2 d d x ∈Z x ∈Z ⎛ ⎞ ⎛ ⎞ 2 2 ⎝ ⎠ ⎝ ⎠ + v (x ) sin p · x + v (x ) sin p · x 1 2 d d x ∈Z x ∈Z 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 625 and v = v + i v for some v , v : Z → R. By Lemma 3.1, the unique eigenvalue 1 2 1 2 e(μ) of H solves f (p)dp 1 − μ = 0. d e(p) − e(μ) d d Since both p ∈ T → f (p) and p ∈ T → f (π + p) are even analytic functions, f ( p)dp we can still apply Propositions A.1 and A.2 to find the expansions of z → T e( p)−z and thus, repeating the same arguments of the proofs of Theorems 2.4 and 2.5 one can obtain the corresponding expansions of e(μ). Remark 3.4 When 2n +d+1 | v(x )|= O(|x | ) as |x|→∞ for some n ≥ 1, in view of Remark A.3, we need to solve equation (3.4) with respect to μ using only that left-hand side is an asymptotic sum (not a convergent series). This still can be done using appropriate modification of the Implicit Function Theorem for differentiable functions. As a result, we obtain only (Taylor-type) asymptotics of e(μ). Acknowledgements Sh. Kholmatov acknowledges support from the Austrian Science Fund (FWF) project M 2571-N32. Funding Open access funding provided by University of Vienna. 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/. Appendix A. Asymptotics of some integrals In this section we study the behaviour of the integral f (q)dq l (z) := , z ∈ C \[0, 4d ], (A.1) e(q) − z 2 d d as z → 0 and z → 4d , where f : T → R is a real-analytic even function on T . Further we denote by W (ξ ) ⊂ C the complex disc of radius r > 0 centered at ξ ∈ C. Proposition A.1 Let f : T → R be a real-analytic even function such that 2 2n −2 2n o o f (0) = D f (0) = ... = D f (0) = 0, D (0) =0(A.2) 123 626 S. Yu. Kholmatov et al. for some n ≥ 0. Then: – l is continuous at 0 if and only if 2n + d ≥ 5; f o – l is continuously differentiable at 0 if and only if 2n + d ≥ 9, in this case, f (q)dq f (q)dq l (0) := = lim . 2 2 d d (e(q)) z0 (e(q) − z) T T Moreover, for any z ∈ (− , 0) : (a) if d is odd, then d n/4 c + a (−z) , 2n + d = 1, 3/4 f o ⎪ 4(−z) n≥1 π d n/4 c + a (−z) , 2n + d = 3, 1/4 f o ⎪ 8(−z) ⎪ n≥1 1/4 π(−z) d n/4 l (z) = l (0) − c + a (−z) , 2n + d = 5, (A.3) f f f o 8 n ⎪ n≥1 3/4 π(−z) d n/4 ⎪ l (0) − c + a (−z) , 2n + d = 7, f f o ⎪ 8 ⎪ n≥1 d n/4 ⎪ l (0) + z l (0) + a (−z) , 2n + d ≥ 9, f o ⎩ f n≥1 (b) if d is even, then π d n/2 1 d n c + b (−z) − ln(−z) c z , 2n + d = 2, 1/2 f o ⎪ n 16 n 8(−z) n≥1 n≥0 d n d n/2 ⎪ − ln(−z) c + c z + b (−z) , 2n + d = 4, f o n n ⎪ 16 ⎪ n≥1 n≥0 1/2 π(−z) d n/2 d n l (z) = l (0) − c + b (−z) + z ln(−z) c z , 2n + d = 6, f f f o n n ⎪ n≥1 n≥0 d n d n/2 ⎪ l (0) − ln(−z) c + c z + b (−z) , 2n + d = 8, f f o n n ⎪ 16 ⎪ n≥1 n≥2 d n/2 2 d n ⎪ l (0) + z l (0) + b (−z) + z ln(−z) c z , 2n + d ≥ 10, f o f n n n≥1 n≥0 (A.4) d d d where {a }, {b } and {c } are some real coefficients, n n n 2n +d 2n d−1 c := D f (0)[w,...,w]dH ; (A.5) d−1 (2n )! o S and all series in (A.3) and (A.4) converge absolutely for z ∈ W (0) ⊂ C. 1/64 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 627 d d Proof Given γ ∈ (0, ], let ϕ : B (0) ⊂ R → ϕ(B (0)) ⊂ R be the smooth γ γ diffeomorphism ϕ (y) = 2 arcsin y , i = 1,..., d. i i Note that 2 2 d d 2 4 e(ϕ(y)) = (1 − cos(2arcsin(y ))) = 4 y = 4y , (A.6) i =1 i =1 therefore, 4 d e(q) ≥ 4γ for any q ∈ T \ ϕ(B ). (A.7) We rewrite l (z) as f (q)dq f (q)dq ∗ ∗∗ l (z) := + := l (z) + l (z). e(q) − z d e(q) − z ϕ(B (0)) T \ϕ(B (0)) γ γ By virtue of (A.7), −1 f (q) z f (q)dq ∗∗ n l (z) = 1 − dq = z , n+1 d e(q) e(q) d (e(q)) T \ϕ(B (0)) T \ϕ(B (0)) γ γ n≥0 (A.8) ∗∗ ∗ i.e. l (·) is analytic in W 4 (0). In l making the change of variables q = ϕ(y) and 2γ using (A.6) we get f (ϕ(y)) J (ϕ(y)) dy l (z) = , (A.9) 4y − z B (0) 4 2 2 2 2 where y := (y ) with y := y , and i =1 J (ϕ(y)) = (A.10) 1 − y i =1 is the Jacobian of ϕ. Since f is an even analytic function satisfying (A.2), even each coordinate, from the Taylor series for f it follows that 2n f (p) = D f (0)[ p,..., p], (A.11) (2n)! n≥n 2n-times 123 628 S. Yu. Kholmatov et al. and by the analyticity of f in B (0) ⊂ R , the series converges absolutely in p ∈ B (0). By the definition of ϕ, ϕ(r w) ⊂ B (0) for any r ∈ (0,γ ) and π π d−1 d−1 d w = (w ,...,w ) ∈ S , where S is the unit sphere in R . Then letting 1 d p = ϕ(r w) and using the Taylor series 3 3 r w i 2n−1 2n−1 ϕ (r w) = 2r w + + c˜ r w i i n n≥3 of 2 arcsin(·), which is absolutely convergent for |r w | < 1, from (A.11) we obtain 2n f (ϕ(r w)) = C (w) r , (A.12) n≥n d−1 d−1 where C : S → R is a homogeneous polynomial of w ∈ S of degree 2n, and ⎡ ⎤ 2n ⎢ ⎥ 2n C (w) = D f (0) w,...,w ⎣ ⎦ (2n )! 2n - times −1/2 Next consider J (ϕ(y)). Inserting the Taylor series of (1 − t ) into (A.10)we obtain ⎛ ⎞ d 2n ⎝ ⎠ J (ϕ(r w)) = 2 1 + C (w)r , (A.13) n≥1 d−1 d−1 where C : S → R is a homogeneous symmetric polynomial of w ∈ S of degree 2n, and the series converges absolutely. Now passing to polar coordinates by y = r w in (A.9) and using (A.12) and (A.13) as well as the absolute convergence of the series we get γ d−1 γ 2n+d−1 r r dr ∗ d 2n d−1 l (z) =2 C (w) r dH dr = c , n n 4 4 4r − z d−1 4r − z 0 S 0 n≥n n≥n o o (A.14) d−1 d−1 where C : S → R is a homogeneous polynomial of w ∈ S of degree 2n and d d−1 c := 2 C (w)dH . n n d−1 Note that c = c , where c is given by (A.5) and the last series in (A.14) uniformly n f f converges in any compact subset of C \[0, 4] since l and γ 2n+d−1 r dr z ∈ C \[0, 4] → j (z) := 2n+d−1 4r − z 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 629 are analytic functions in C \[0, 4] and all series in (A.14) converge pointwise .Note that for any m ≥ 0, there exist c ∈ R and an analytic function f in the ball m m W 4 (0) ⊂ C such that for any z ∈ (−γ , 0), n o ν 1/2 j (z) = z j (z) + c + z f ((−z) ), (A.15) m m m m 1 where n := [ ], l := m − 4n ∈{0, 1, 2, 3},ν = for m = 0, 2 and ν = 1for 4 2 m = 1,3or m ≥ 4, and −3/4 (−z) if l = 0, ⎪ 4 ⎨ π −1/2 (−z) if l = 1, j (z) := π −1/4 (−z) if l = 2, − ln(−z) if l = 3. Inserting (A.15)into(A.14) we obtain 2n+d−1 ∗ [ ] o l (z) = c z j (z) 2n+d−1 2n+d−1−4[ ] n≥n ν 1/2 +c + c (−z) f ((−z) ) , 2n+d−1 n 2n+d−1 where {c }⊂ R and { f } is a sequence of analytic functions in W 4 (0) 2n+d−1 2n+d−1 and , 2n + d = 1, 3, ν := 1, otherwise. Since (A.14) converges locally uniformly in C \[0, 4], C := c c is finite n 2n+d−1 n≥n and ν 1/2 ν 1/2 c (−z) f ((−z) ) = (−z) g((−z) ), n 2n+d−1 n≥n where g is analytic in W 2 (0) and ν = for 2n + d = 1, 3 and ν = 1 otherwise. γ o Hence, 2n+d−1 ∗ ν 1/2 [ ] o l (z) = C + (−z) g((−z) ) + c z j (z), (A.16) 2n+d−1 2n+d−1−4[ ] n≥n If {h } is an equi-bounded sequence of analytic functions in a connected open set ⊂ C converging pointwise to a function h : → C, then h is analytic and h converges uniformly to h in compact subsets of . 123 630 S. Yu. Kholmatov et al. If 0 ≤ 2n + d − 1 ≤ 3, then by (A.16), ∗ ν 1/2 o l (z) = C + (−z) g((−z) ) + c j (z) o 2n +d−1 2n+d−1 [ ] o + c z j (z). (A.17) n 2n+d−1 2n+d−1−4[ ] n≥n +1 In view of (A.8) and the definition of j , from (A.17) we obtain the expansions (A.3) 2n+d−1 and (A.4)of l for 2n + d ≤ 4. In particular, since [ ]≥ 1for n ≥ n + 1, f o o letting z → 0in (A.17) we get lim l (z) =+∞. (A.18) z→0 2n+d−1 If 2n + d − 1 ≥ 4, then [ ]≥ 1 for any n ≥ n . Therefore, by (A.16), o o ∗ ∗ l (0) := lim l (z) exists and equals to C . In particular, for 2n + d − 1 ≤ 7, one has z→0 ∗ ∗ 1/2 o l (z) = l (0) − zg((−z) ) + c zj (z) o 2n +d−1 2n+d−1 [ ] o + c z j (z), (A.19) 2n+d−1 2n+d−1−4[ ] n≥n +1 from which and (A.8) we deduce the expansions (A.3) and (A.4)of l for 5 ≤ 2n +d ≤ f o ∗∗ 8. In particular, by virtue of (A.18) and analyticity of l at z = 0, l is continous at 0 if and only if 2n + d ≥ 5. Notice also by (A.19) ∗ ∗ l (z) − l (0) lim =+∞, (A.20) z→0 z i.e. l (and hence l ) is not differentiable at z = 0. 2n+d−1 Finally, if 2n + d − 1 ≥ 8, then [ ]≥ 2 for any n ≥ n . Therefore, by o o (A.16) there exists ∗ ∗ l (z) − l (0) l (0) := lim =−g(0). z→0 z Now using the Taylor series of g at 0 we get (n) g (0) 1/2 ∗ n/2 zg((−z) ) = l (0)z + z (−z) . n! n≥1 ∗∗ Inserting this in (A.16), using the definition of j and the analyticity of l we get the expansions (A.3) and (A.4)of l for 2n + d ≥ 9. f o By (A.18) and (A.20), l is continously differentiable at 0 if and only if 2n +d ≥ 9. f o Now the choice γ = completes the proof. 123 Expansion of eigenvalues of the perturbed discrete bilaplacian 631 d d Proposition A.2 Let f : T → R be a real-analytic function such that q ∈ T → f (π + q) is even and 2 2n −2 2n o o f (π) = D f (π) = ... = D f (π) = 0, D (π) = 0 for some n ∈ N . Then: o 0 – l is continuous at z = 4d if and only if for 2n + d ≥ 3, f o – l is continuously differentiable at z = 4d if and only if for 2n + d ≥ 5, in this f o case f (q)dq f (q)dq l (4d ) := = lim 2 2 2 d d (e(q) − 4d ) (e(q) − z) z4d T T exists. 2 1 Moreover, if z − 4d ∈ (0, ), l (z) is represented as: (a) if d is odd, then πC d 2 k/2 ⎪ √ − + a (z − 4d ) , 2n + d = 1, ⎪ o 2 k z−4d k≥0 ⎨ √ 2 d 2 k/2 l (4d ) + πC z − 4d + a (z − 4d ) , 2n + d = 3, f f o l (z) = k ⎪ k≥2 ⎪ 2 2 2 d 2 k/2 l (4d ) + l (4d )(z − 4d ) + a (z − 4d ) , 2n + d ≥ 5; f o f k k≥3 (A.21) (b) if d is even, then d d k k C ln α + ln α b α + c α , 2n + d = 2, f o k k ⎪ k≥1 k≥0 2 d k d k l (4d ) − C α ln α + ln α b α + c α , 2n + d = 4, f f o l (z) = k k ⎪ k≥2 k≥1 d d ⎪ 2 2 k k ⎩ l (4d ) + l (4d )α + ln α b α + c α , 2n + d ≥ 6, f o f k k k≥2 k≥2 (A.22) 2 d d d where α := z − 4d , {a }, {b }, {c }⊂ R and k k k 2n +d−1 2n d−1 C := D f (π) [w,...,w]dH . n +d/2 d−1 (8d) (2n )! o S Proof Since 4d − e(·) has a unique non-degenerate minimum at π, the asymptotics of l (z) as z 4d can be done along the lines of, for instance, [22,Lemma 4.1], hence, we skip the proof. 123 632 S. 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Monatshefte für Mathematik – Springer Journals
Published: Apr 1, 2022
Keywords: Discrete bilaplacian; Essential spectrum; Discrete spectrum; Eigenvalues; Asymptotics; Expansion; 47A10; 47A55; 47A75; 41A60
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