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Some generic properties of Schrödinger operators with radial potentials

Published online by Cambridge University Press:  15 January 2019

Peter Poláčik
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA ([email protected])
Darío A. Valdebenito
Affiliation:
Department of Mathematics and Statistics, McMaster University Hamilton, ON L8S 4K1, Canada ([email protected])

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019 

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