## Abstract

We prove compactness of a restricted set of real-valued, compactly supported potentials V for which the corresponding Schrödinger operators HV have the same resonances, including multiplicities. More specifically, let BR(0) be the ball of radius R > 0 about the origin in Rd, for d = 1, 3. Let IR(V0) be the set of real-valued potentials in C∞ 0 (BR(0);R) so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as HV0 , for a fixed, but arbitrary, potential V0 ∈ C∞ 0 (BR(0);R). We prove that the set IR(V0) is a compact subset of C∞ 0 (BR(0)) in the C∞-topology. An extension to Sobolev spaces of less regular potentials is discussed.

Original language | English |
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Pages (from-to) | 4481-4499 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 374 |

Issue number | 6 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Publisher Copyright:© 2021 American Mathematical Society. All rights reserved.

## Keywords

- Isopolar potentials
- Resonances
- Schrödinger operators

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics