Orbital Stability for the Schr\"{o}dinger Operator Involving Inverse Square Potential

TL;DR

This paper establishes the existence of orbitally stable standing waves for a critical Schrödinger operator with an inverse square potential using variational methods, and explores effects of Hardy energy terms near singularities.

Contribution

It provides a novel variational proof of orbital stability for Schrödinger operators with inverse square potentials, including cases with Hardy energy terms.

Findings

Existence of orbitally stable standing waves proven

Precompactness of minimizing sequences demonstrated

Analysis of behavior near singularities conducted

Abstract

In this paper we prove the existence of orbitally stable standing waves for the critical Schr\"{o}dinger operator, involving potential of the form $\left(\frac{N-2}{2}\right)^2|x|^{-2}$. The approach, being purely variational, is based on the precompactness of any minimizing sequence with respect to the associated energy. Moreover, we discuss the case of the presence of a Hardy energy term, in conjunction with the behavior of the standing waves at the singularity.