On the spectral instability of the Sturm-Liouville operator with a complex potential
type: Item
instance of: scholarly article
title: On the spectral instability of the Sturm-Liouville operator with a complex potential
zbMATH Open document ID: 1188.47037
DOI: 10.1134/S001226610904003X
author: Kh. K. Ishkin
published in: Differential Equations
publication date: 2009-10-08T00:00:00Z
review text: For selfadjoint operators, the spectrum of a perturbation can be estimated by relative bounds of the perturbations. However, this is no longer true for non-selfadjoint operators, and it is therefore interesting to find necessary and sufficient conditions to preserve spectral asymptotics, or to find counterexamples. In this paper, the author is concerned with the case that the spectrum is located around a ray. In Theorem 1, sufficient conditions are given to ensure that if the spectra of the operator \(T_0\) and its perturbation \(T_0+V\) both are located around a ray, say \(\arg \lambda =\varphi_0 \) and \(\arg \lambda = \psi _0\), respectively, then \(\psi _0 =\varphi_0 \). This result is then applied to the anharmonic oscillator \(H_\theta \) in \(L^2(0,\infty )\) given by \(-y''+e^{i\theta }x^\alpha y\) (\(\alpha >0\), \(0<|\theta |<\pi\)) and the boundary condition \(y(0)=0\). \(H_\theta \) is perturbed by a potential of the form \(V_0+V\), where the assumptions on \(V_0\) imply that \(H_\theta \) and \(H_\theta +V_0\) have the same spectral asymptotics. However, if \(V\neq0\), then \(V\) is assumed to be of the form \(V(x)=(b-x)^mW(x)\), where \(m\geq0\), supp\,\(W=[0,b]\) and \(W(b-0)\neq0\) exists, and it follows that \(H_\theta +V_0+V\) has two eigenvalue sequences, one with the same asymptotics as \(H_\theta \) and the second one having the same asymptotics as the squares of the eigenvalues of the Regge problem on \([0,b]\). This shows that a small change \(W\) of the potential does not lead to a small change in the spectrum.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 5611056
zbMATH Keywords: spectral asymptotics
zbMATH Keywords: anharmonic oscillator
zbMATH Keywords: Sturm-Liouville problem
zbMATH Keywords: Regge problem
describes a project that uses: Eigtool
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5611056