Compactness of iso-resonant potentials for schrödinger operators in dimensions one and three

PETER D. HISLOP, ROBERT WOLF

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1 Scopus citations

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 languageEnglish
Pages (from-to)4481-4499
Number of pages19
JournalTransactions of the American Mathematical Society
Volume374
Issue number6
DOIs
StatePublished - 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

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