Spectral rigidity for radial Schrödinger operators
type: Item
instance of: scholarly article
title: Spectral rigidity for radial Schrödinger operators
zbMATH Open document ID: 0809.34088
author: V. Pereyra
published in: Journal of Differential Equations
publication date: 1995-03-30T00:00:00Z
review text: The authors study the problem of identifying the spectral data needed to determine the coefficients of the radial Schrödinger operators \(- \Delta+ p(\| x\|)\) in odd space dimensions \(n\) greater than 1 acting on functions on the unit ball with a boundary condition such as \(f(x)= 0\) when \(\| x\|= 1\) or \(\partial f/\partial r(x)+ bf(x)= 0\) when \(\| x\|= 1\). The eigenvalues of the operator can be obtained as the union of the eigenvalues of a sequence of singular ordinary differential operators \(-D^ 2+ p(x)+ m(m+1)/ x^ 2\) where \(m= m_ 0\), \(m+1\), \(m+2,\dots\) with \(m_ 0= (n- 3)/2\). A local diffeomorphism taking operators \(S(m,b,p)\) to those of the form \(S(m+1,b,p)\) is analyzed, the derivative of this diffeomorphism is computed, and these linear operators are related to convolution operators on the line. Finally it is proved that if \(m(0)+ m(1)= 1\text{ mod }2\) then the tangent spaces \(\text{TM}(m(0),b,Q)\) and \(\text{TM}(m(1),b,Q)\) are Fredholm for any \(Q\in L^ 2_ R[0,1]\). Fredholm operators and subspaces are discussed in the Appendix.
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 665704
zbMATH Keywords: spectral data
zbMATH Keywords: radial Schrödinger operators
zbMATH Keywords: eigenvalues
zbMATH Keywords: convolution operators
zbMATH Keywords: Fredholm operators
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1127
OpenAlex ID: W1964708204
DOI: 10.1006/JDEQ.1994.1127
description: scientific article; zbMATH DE number 665704