Spectrum and pseudospectrum of non-self-adjoint Schrödinger operators with periodic coefficients
type: Item
instance of: scholarly article
title: Spectrum and pseudospectrum of non-self-adjoint Schrödinger operators with periodic coefficients
zbMATH Open document ID: 1128.47042
author: S. V. Gal'tsev
author: A. I. Shafarevich
published in: Mathematical Notes
publication date: 2007-05-21T00:00:00Z
review text: The authors consider the pseudospectrum of the non-selfadjoint operator \[ D=-h^2\frac{d^2}{dx^2}+iV(x), \] where \(V(x)\) is an aperiodic entire analytic functions real on the real axis, with a real period \(T\). The operator \(h\) is treated as a small parameter. The authors show that the pseudospectrum of this operator is the closure of its numerical image, i.e., a half-strip in \(\mathbb C\). In this case, the pseudo-eigenfunctions, i.e., the functions \(\varphi(h,x)\) satisfying the condition \[ \|D\varphi-\lambda\varphi\|=O(h^n),\quad \|\varphi\|= 1,\;n\in\mathbb N, \] can be constructed explicitly. The main result of the paper is as follows. Theorem. For a non-constant \(V(x)\) and for any positive integer \(n\), the \(h^n\)-pseudo-spectrum of the operator \(-h^2\frac{d^2} {dx^2}+iV(x)\), given on a circle, is the half-strip \([0,\infty)+i[\min V, \max V]\). Thus, it turns out that the pseudospectrum of the operator under study is much wider than its spectrum.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5155533
zbMATH Keywords: spectrum
zbMATH Keywords: pseudospectrum
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: periodicity condition
zbMATH Keywords: periodic entire analytic function
zbMATH Keywords: non-selfadjoint operator
zbMATH Keywords: Riemann surface
reviewed by: John Schmeelk
MaRDI profile type: Publication
full work available at URL: s11006-006-0146-z
OpenAlex ID: W2035640823
cites work: Q5688841
cites work: Asymptotically stable invariant tori of a vector field \(V(x)\) and the quasimodes of the operators \(V(x) \cdot \nabla- \varepsilon \Delta\)
cites work: Non-selfadjoint singular perturbations and spectral properties of the Orr-Sommerfeld boundary-value problem
cites work: The limit behavior of the spectrum for large parameter values in a model problem
cites work: On localization of the spectrum in a problem in singular perturbation theory
cites work: On the limit behaviour of the spectrum of a model problem for the Orr-Sommerfeld equation with Poiseuille profile
DOI: 10.1007/S11006-006-0146-Z
description: scientific article; zbMATH DE number 5155533