Construction of self-adjoint differential operators with prescribed spectral properties

Jussi Behrndt*, Andrii Khrabustovskyi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

Original languageEnglish
Pages (from-to)1063-1095
Number of pages33
JournalMathematische Nachrichten
Issue number6
Publication statusPublished - Jun 2022


  • boundary condition
  • differential operator
  • discrete spectrum
  • essential spectrum
  • Neumann Laplacian
  • Schrödinger operator
  • singular potential

ASJC Scopus subject areas

  • Mathematics(all)


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