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

Abstract

Simon’s results on the negative spectrum of recurrent Schrödinger operators ( d = 1 , 2 d=1,2 ) are extended to a wider class of potentials and to non-local operators. An example of L 1 − L^1- potental is constructed for which the essential spectrum of two dimensional Schrödinger operator covers the whole axis. Some counterexamples are provided for transient operators ( d ≥ 3 d\geq 3 ) showing that the assumptions on the potential for the validity of the Cwikel-Lieb-Rozenblum estimate can’t be improved significantly.