An inverse spectral problem for a nonsymmetric differential operator: reconstruction of eigenvalue problem
type: Item
instance of: scholarly article
title: An inverse spectral problem for a nonsymmetric differential operator: reconstruction of eigenvalue problem
zbMATH Open document ID: 1114.34013
author: Wuqing Ning
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: The paper deals with the non-selfadjoint boundary value problem \[ B\mathbf v'(x)+P(x)\mathbf v(x)=\lambda\mathbf v(x),\quad 0<x<1, \] \[ \mathbf v'_2(0)\cos\mu-\mathbf v_1(0)\sin\mu=0,\; \mathbf v'_2(1)\cos\nu+\mathbf v_1(1)\sin\nu=0, \] where \(\lambda\) is the spectral parameter, \[ \mathbf v(x)= \left[ \begin{matrix} \mathbf v_{1}(x) \\ \mathbf v_{2}(x) \end{matrix} \right], \; B= \left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right],\; P(x)= \left[ \begin{matrix} p_{11}(x) & p_{12}(x) \\ p_{21}(x) & p_{22}(x) \end{matrix} \right], \] \(p_{jk}(x)\in C^1[0,1]\) are complex-valued functions, and \(\mu,\nu\) are complex numbers. The author studies the following inverse problem: Let \(p_{21}(x), p_{22}(x)\) and \(\mu\) are known a priori and fixed. Given the spectral data \(S,\) construct \(p_{11}(x), p_{12}(x)\) and \(\nu\). Using the transformation operator method (which is due to Marchenko and Levitan) the author provides a constructive procedure for the solution of this inverse problem along with necessary and sufficient conditions for its solvability.
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 5083359
zbMATH Keywords: non-self-adjoint differential operators
reviewed by: V. A. Yurko
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.05.015
OpenAlex ID: W2051169841
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DOI: 10.1016/J.JMAA.2006.05.015
description: scientific article; zbMATH DE number 5083359