Spectral shift via 'lateral' perturbation
| Autor: | Gregory Berkolaiko, Peter Kuchment |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Journal of Spectral Theory. 12:83-104 |
| ISSN: | 1664-039X |
| DOI: | 10.4171/jst/395 |
| Popis: | We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $\lambda^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be "along" the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $\lambda^\circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $\sigma$ more eigenvalues below $\lambda^\circ$ than $H_0$; $\sigma$ is known as the spectral shift at $\lambda^\circ$. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $\lambda^\circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $\sigma$. A version of this theorem also holds for some non-positive perturbations. Comment: 18 pages, 2 figures; dedicated to memory of Misha Shubin revised following referee suggestions; several references added |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|