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


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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
Issue number6
StatePublished - 2021

Bibliographical note

Funding Information:
Received by the editors March 14, 2019, and, in revised form, August 27, 2020, and November 29, 2020. 2020 Mathematics Subject Classification. Primary 35R01, 58J50, 47A55. Key words and phrases. Resonances, Schrödinger operators, isopolar potentials. Both authors were partially supported by NSF grant DMS 11-03104 during the time this work was done. This paper is partly based on the dissertation submitted by the second author in partial fulfillment of the requirements for a PhD at the University of Kentucky.

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


  • Isopolar potentials
  • Resonances
  • Schrödinger operators

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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