## Abstract

We consider a class of Schrödinger operators on ℝ^{N}with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in L^{2}(ℝ^{N}), 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 language | English (US) |
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Pages (from-to) | 1435-1451 |

Number of pages | 17 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 149 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2019 |

### Bibliographical note

Funding Information:Supported in part by the NSF Grant DMS-1565388. CONICYT-Chile Becas Chile, Convocatoria 2010

Funding Information:

Supported in part by the NSF Grant DMS-1565388. Supported in part by CONICYT-Chile Becas Chile, Convocatoria 2010

Publisher Copyright:

Copyright © Royal Society of Edinburgh 2019.

## Keywords

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