Negative eigenvalues of non-local Schrödinger operators with sign-changing potentials

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 $L<sup>1-$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.