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Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients In the space L 2[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L 2[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ n=1 ∞ [µ n − λ n − Σ k=1 m α k (n) ] = 0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

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References (2)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Pleiades Publishing, Ltd.
Subject
Mathematics; Difference and Functional Equations; Partial Differential Equations; Ordinary Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266108120057
Publisher site
See Article on Publisher Site

Abstract

In the space L 2[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L 2[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ n=1 ∞ [µ n − λ n − Σ k=1 m α k (n) ] = 0.

Journal

Differential Equations – Springer Journals

Published: Feb 11, 2009

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