Some generic properties of Schrödinger operators with radial potentials

Peter Poláčik, Darío A. Valdebenito

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a class of Schrödinger operators on ℝNwith radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in L2(ℝN), we show that the following properties are generic with respect to the potential: (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;(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.

Original languageEnglish (US)
Pages (from-to)1435-1451
Number of pages17
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume149
Issue number6
DOIs
StatePublished - Dec 1 2019

Keywords

  • Schrödinger operators
  • algebraic independence of eigenfunctions
  • generic properties
  • radial potentials
  • rational independence of eigenvalues

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