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Some generic properties of Schrödinger operators with radial potentials

Part of: Elliptic equations and systems General theory of linear operators Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  15 January 2019

Peter Poláčik  and
Darío A. Valdebenito
Peter Poláčik
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA (polacik@math.umn.edu)
Darío A. Valdebenito
Affiliation:
Department of Mathematics and Statistics, McMaster University Hamilton, ON L8S 4K1, Canada (valdebed@math.mcmaster.ca)
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Abstract

We consider a class of Schrödinger operators on ${\open R}^N$ with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in $L^2({\open R}^N)$, we show that the following properties are generic with respect to the potential:

  1. (P1) the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;

  2. (P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set.

The genericity means that in suitable topologies the potentials having the above properties form a residual set. As we explain, (P1), (P2) are prerequisites for some applications of KAM-type results to nonlinear elliptic equations. Similar properties also play a role in optimal control and other problems in linear and nonlinear partial differential equations.

Keywords

Schrödinger operatorsradial potentialsrational independence of eigenvaluesalgebraic independence of eigenfunctionsgeneric properties

MSC classification

Primary: 35J10: Schrödinger operator
Secondary: 35P99: None of the above, but in this section 47A75: Eigenvalue problems 47A55: Perturbation theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1435 - 1451
Copyright
Copyright © Royal Society of Edinburgh 2019 

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