Spectral analysis of a class of second-order non-self-adjoint differential operators
type: Item
instance of: scholarly article
title: Spectral analysis of a class of second-order non-self-adjoint differential operators
zbMATH Open document ID: 0574.34012
DOI: 10.1007/BF01078408
author: M. G. Gasymov
publication date: 1980-01-01T00:00:00Z
review text: In this article, we study the spectrum and the eigenfunction expansion of a differential operator L, generated by the differential expression \(\ell (y)\equiv -y''+q(x)y\) in the space \(L_ 2(-\infty,\infty)\), under the assumption that the coefficient is \(q(x)=\sum^{\infty}_{n=1}q_ ne^{inx}\) and the series \(\sum^{\infty}_{n=1}| q_ n| =q\) converges. Clearly q(x) is a periodic function on (-\(\infty,\infty)\) (the period is equal to \(2\pi)\), and it admits holomorphic continuation to the upper half plane. Therefore, the operator L is not self-adjoint, and becomes self-adjoint only in the trivial case, for q(x)\(\equiv 0\). It turns out that the spectrum of L is purely continuous, fills the semiaxis [0,\(\infty)\), and there may be second-order spectral singularities on the continuous spectrum, which must coincide with numbers of the form \((n/2)^ 2\). For the generalized eigenfunctions corresponding to the spectral singularities, we may introduce the idea of generalized norming numbers \(\{s_ n\}\). We prove that from these we may effectively reconstruct \(\{q_ n\}\).
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 3916759
zbMATH Keywords: spectrum
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: spectral singularities
MaRDI profile type: Publication
cites work: Q5682632
cites work: The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
full work available at URL: https://doi.org/10.1007/bf01078408
OpenAlex ID: W1977227789
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3916759