Bounds of Dirichlet eigenvalues for Hardy-Leray operator

TL;DR

This paper establishes bounds and asymptotic behavior of eigenvalues for the Dirichlet Hardy-Leray operator, revealing that the inverse-square potential does not affect the eigenvalue growth rate.

Contribution

It provides new lower and upper bounds for the eigenvalues of the Hardy-Leray operator and shows the Weyl limit is independent of the potential parameter.

Findings

Lower bounds for eigenvalues including Li-Yau and Karachalio bounds.

Cheng-Yang type upper bounds for eigenvalues.

Eigenvalue asymptotics are unaffected by the inverse-square potential.

Abstract

The purpose of this paper is to study the eigenvalues $\{\lambda_{\mu,i} \}_i$ for the Dirichlet Hardy-Leray operator, i.e. $$ -\Delta u+\mu|x|^{-2}u=\lambda u\ \ {\rm in}\ \, \Omega,\quad\quad u=0\ \ {\rm on}\ \ \partial\Omega,$$ where $-\Delta +\frac{\mu}{|x|^2}$ is the Hardy-Leray operator with $\mu\geq -\frac{(N-2)^2}{4}$ and $\Omega$ is a smooth bounded domain with $0\in\Omega$. We provide lower bounds of $\{\lambda_{\mu,i} \}_i$ together with the Li-Yau's one for $\mu>-\frac{(N-2)^2}{4}$ and Karachalio's one for $\mu\in [-\frac{(N-2)^2}{4},0)$. Secondly, we obtain Cheng-Yang's type upper bounds for $\lambda_{\mu,k}$. Finally, we get the Weyl's limit of eigenvalues which is independent of the potential's parameter $\mu$. This interesting phenomena indicates that the inverse-square potential does not play an essential role for the asymptotic behavior of the spectral of the problem considered.