Ground State for the Schrödinger Operator with the Weighted Hardy Potential

J Chabrowski,K Tintarev

doi:10.1155/2011/358087

Abstract

We establish the existence of ground states on ℝN 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.

Highlights

We investigate the existence of ground states of the Schrodinger operator associated with the quadratic form
Where V belongs to the Lorentz space LN/2, ∞ RN and ΛV is the largest constant whenever exists for which the form QV is nonnegative. This assumption implies, when V ≥ 0, that the potential term RN V x u2 dx is continuous in D1,2 RN, where D1,2 RN is the Sobolev space obtained as the completion of C◦∞ RN with respect to the norm u2
We examine the behaviour of the principal eigenfunction around 0

Summary

Introduction

We investigate the existence of ground states of the Schrodinger operator associated with the quadratic form. If QV admits a null sequence vk, the sequence vk converges weakly in Hl1oc RN to a unique (up to a multiplicative constant) positive solution of 1.4 This theorem gives rise to the definition of the generalized ground state. We mention papers 2, 6–13 , where the existence of principal eigenfunctions has been established under various assumptions on weight functions These conditions require that a potential belongs to some Lebesgue space, for example Lp RN with p > N/2. In paper 15 the existence of principal eigenfunctions has been established for weights belonging to 1≤q

Existence of Minimizers

Perturbations from Virtual Ground States

Behaviour of a Ground State Around 0