A Schr\"odinger operator with confining potential having quadratic growth

TL;DR

This paper investigates the spectral properties of a Schrödinger operator with a quadratic confining potential, providing continuity estimates, bounds, regularity, and decay properties of eigenvalues and eigenstates.

Contribution

It offers new continuity estimates, bounds, and decay properties for eigenvalues and eigenstates of Schrödinger operators with quadratic confining potentials.

Findings

Continuity estimates for eigenvalues and eigenstates

Lower bounds on ground state energy

Explicit decay estimates at infinity

Abstract

We study the spectral properties of a Schr\"odinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates, lower bounds on the ground state energy, regularity and integrability properties of eigenstates. We also get explicit decay estimates at infinity, by means of elementary nonlinear methods.