Inverse spectral problems for Sturm--Liouville operators in impedance form
type: Item
instance of: scholarly article
title: Inverse spectral problems for Sturm--Liouville operators in impedance form
zbMATH Open document ID: 1082.34006
published in: Journal of Functional Analysis
publication date: 2005-06-01T00:00:00Z
review text: The paper presents a study of the direct and inverse spectral problems for a class of Sturm-Liouville equations on an interval in the so-called impedance form \[ -(a^2(x)y'(x))'=\lambda^2 a^2(x)y(x), \quad x\in[0,1], \] subjected to certain boundary conditions, namely, the Neumann boundary conditions \(y'(0)=y'(1)=0\) and the Neumann--Dirichlet ones \(y'(0)=y(1)=0\). The impedance function \(a(x)\) is assumed to be positive and to belong to the Sobolev space \(W_p^1(0,1)\) for some \(p\in[1,\infty)\). The presented approach is actually aimed at recovery of the logarithmic derivative \(\tau=(\log a)'\in L_p(0,1)\). The authors give a complete solution of the direct and inverse problems for an arbitrary \(p\in[1,\infty)\), i.e., they give a complete description of all possible spectra for the class of operators under consideration (basically, it is the description of the asymptotics of the eigenvalues) and for any such spectral data, they provide an effective algorithm of recovery of the function \(\tau\).
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34L20
zbMATH DE Number: 2172990
zbMATH Keywords: inverse spectral problem
zbMATH Keywords: Sturm-Liouville operators
zbMATH Keywords: impedance form
author: S. Albeverio
author: R. Hryniv
author: Ya. Mykytyuk
reviewed by: Dmitry Shepelsky
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.08.010
OpenAlex ID: W2086343131
cites work: Singular Perturbations and Nonstandard Analysis
cites work: Q3742397
cites work: Q3994511
cites work: Q4938042
cites work: Inverse eigenvalue problems with discontinuous coefficients
cites work: Inverse eigenvalue problems for a Sturm-Liouville equation in impedance form
cites work: Multipliers in dual Sobolev spaces and Schrödinger operators with distribution potentials
cites work: Schrödinger operators with singular interactions
cites work: Singular Schrödinger operators as limits of point interaction Hamiltonians
cites work: Solution of the inverse spectral problem for an impedance with integrable derivative part I
cites work: Q3266407
cites work: On the determination of differential equations with singularities and turning points
cites work: Q5535688
cites work: Q5553587
cites work: Classes of linear operators. Vol. II
cites work: Q5512384
cites work: Q5590313
cites work: Inverse problems: recovery of BV coefficients from nodes
cites work: Q4171170
cites work: Riesz bases of solutions of Sturm-Liouville equations
cites work: Q3351786
cites work: 1-D Schrödinger operators with periodic singular potentials
cites work: Q4780934
cites work: Inverse spectral problems for Sturm Liouville operators with singular potentials
cites work: Q3374340
cites work: Inverse spectral problems for Sturm--Liouville operators with singular potentials. III: Reconstruction by three spectra.
cites work: Half-inverse spectral problems for Sturm–Liouville operators with singular potentials
cites work: Transformation operators for Sturm--Liouville operators with singular potentials
cites work: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials
cites work: Q4821311
cites work: Q5827231
cites work: Spectral asymptotics for Schrödinger operators with periodic point interactions
cites work: Q3695595
cites work: Q4183612
cites work: Factorization of Fredholm operators.
cites work: Factorization of Fredholm operators in operator algebras.
cites work: Schrödinger operators with singular potentials from the space of multiplicators
cites work: Q3692084
cites work: Fundamental theorems for linear measure differential equations.
cites work: The reconstruction of Sturm-Liouville operators
cites work: On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential
cites work: Sturm-Liouville operators with singular potentials
cites work: Q4656592
DOI: 10.1016/J.JFA.2004.08.010
description: scientific article; zbMATH DE number 2172990