Spectral shift via “lateral” perturbation

  • Gregory Berkolaiko

    Texas A&M University, College Station, USA
  • Peter Kuchment

    Texas A&M University, College Station, USA
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We consider a compact perturbation of a self-adjoint operator with an eigenvalue below its essential spectrum and the corresponding eigenfunction . The perturbation is assumed to be “along” the eigenfunction , namely . The eigenvalue belongs to the spectra of both and . Let have more eigenvalues below than ; is known as the spectral shift at .

We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue in the spectrum of . We show that the eigenvalue as a function of has a critical point at and the Morse index of this critical point is the spectral shift . A version of this theorem also holds for some non-positive perturbations.

Cite this article

Gregory Berkolaiko, Peter Kuchment, Spectral shift via “lateral” perturbation. J. Spectr. Theory 12 (2022), no. 1, pp. 83–104

DOI 10.4171/JST/395