Ground state for the Schr\"odinger operater with the weighted Hardy potential

TL;DR

This paper proves the existence of ground states and principal eigenfunctions for the Laplace operator with weighted Hardy potentials on Euclidean space, and analyzes their behavior near zero.

Contribution

It establishes the existence of ground states and principal eigenfunctions for Laplace operators with weighted Hardy potentials, including their local behavior.

Findings

Existence of ground states for Laplace with Hardy potentials

Higher integrability of principal eigenfunctions

Behavior of eigenfunctions near zero

Abstract

We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.