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

TL;DR

This paper studies the stability of eigenvalues of non-local Schr"odinger operators under potential variations and demonstrates that ground states in certain relativistic models are radially decreasing.

Contribution

It provides new results on the spectral stability of non-local Schr"odinger operators and establishes the radial symmetry of ground states in relativistic quantum models.

Findings

Spectral stability under potential convergence conditions.

Strong and norm resolvent convergence proofs.

Ground states are radially decreasing functions.

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.