Stability of ground state eigenvalues of non-local Schr\"odinger operators with respect to potentials and applications

Abstract

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schr\"odinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the sequence of potentials, we give either direct proofs to show the strong or norm resolvent convergence of the so-obtained sequence of non-local Schr\"odinger operators, or via $\Gamma$-convergence of the related positive forms for more rough potentials. In a second part we use these results to show via a sequence of suitably constructed approximants that the ground states of massive or massless relativistic Schr\"odinger operators with spherical potential wells are radially decreasing functions.