Negative eigenvalues of non-local Schr\"{o}dinger operators with sign-changing potentials

TL;DR

This paper extends results on negative spectra of Schr"{o}dinger operators to non-local cases with sign-changing potentials, providing new examples and counterexamples that challenge existing spectral estimates.

Contribution

It broadens the class of potentials and operators for which spectral properties are understood, including non-local and sign-changing potentials.

Findings

Constructed an $L^1$-potential with the essential spectrum covering the entire axis.

Extended Simon's results to non-local Schr"{o}dinger operators.

Provided counterexamples showing limitations of spectral estimates for transient operators.

Abstract

Simon's results on the negative spectrum of recurrent Schr\"{o}dinger operators ($d=1,2$) are extended to a wider class of potentials and to non-local operators. An example of $L^1-$potental is constructed for which the essential spectrum of two-dimensional Schr\"{o}dinger operator covers the whole axis. Some counterexamples are provided for transient operators ($d\geq3$) showing that the assumptions on the potential for the validity of the Cwikel-Lieb-Rozenblum estimate can't be improved significantly.