Inverse problems on the least eigenvalue
type: Item
instance of: scholarly article
title: Inverse problems on the least eigenvalue
zbMATH Open document ID: 1319.34029
author: Jie Yang
author: Chuan-Fu Yang
published in: Results in Mathematics
publication date: 2014-06-18T00:00:00Z
review text: The paper deals with generalizations of the classical Ambarzumyan theorem, which, as is well-known, can be formulated in the following way: if the first (i.e., smallest) eigenvalue \(\lambda_0\) of the Sturm-Liouville operator \[ Ay:=-y''+q(x)y, \quad 0<x<\pi, \quad y'(0)=y'(\pi)=0 \] with a real-valued potential \(q(x)\in L(0,\pi)\) equals to the mean value of the potential, i.e. \[ \lambda_0=\frac1\pi\int_0^\pi q(x)\,dx, \] then \(q(x)=\lambda_0\) a.e. on \((0,\pi).\) Using the variational principle the authors obtain an analogous result for bounded from below operators \(A\) in an abstract separable Hilbert space. As a corollary they obtain the corresponding facts for selfadjoint second-order differential operators with operator-valued potentials, higher-order partial differential operators, etc.
reviewed by: Sergey Buterin
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 6305234
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: the least eigenvalue
zbMATH Keywords: uniqueness theorem
zbMATH Keywords: Ambarzumyan theorem
zbMATH Keywords: variational principle
MaRDI profile type: Publication
full work available at URL: s00025-013-0347-1
OpenAlex ID: W2056888708
cites work: Sturm--Liouville theory for the p-Laplacian
cites work: Q4530224
cites work: On the regularized trace of a second order differential operator
cites work: Q4109839
cites work: The inverse nodal problem and the Ambarzumyan problem for the p-Laplacian
cites work: On Ambarzumyan-type theorems
DOI: 10.1007/S00025-013-0347-1
description: scientific article; zbMATH DE number 6305234