JournalsjstVol. 12, No. 1pp. 83–104

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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Abstract

We consider a compact perturbation H0=S+K0K0H_0 = S + K_0^* K_0 of a self-adjoint operator SS with an eigenvalue λ\lambda^\circ below its essential spectrum and the corresponding eigenfunction ff. The perturbation is assumed to be “along” the eigenfunction ff, namely K0f=0K_0f=0. The eigenvalue λ\lambda^\circ belongs to the spectra of both H0H_0 and SS. Let SS have σ\sigma more eigenvalues below λ\lambda^\circ than H0H_0; σ\sigma is known as the spectral shift at λ\lambda^\circ.

We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue λ\lambda^\circ in the spectrum of H(K)=S+KKH(K)=S + K^* K. We show that the eigenvalue as a function of KK has a critical point at K=K0K=K_0 and the Morse index of this critical point is the spectral shift σ\sigma. 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