Bounds of Dirichlet eigenvalues for Hardy-Leray operator

Abstract

The purpose of this paper is to study the eigenvalues $\{λ_{μ,i} \}_i$ for the Dirichlet Hardy-Leray operator, i.e. $$ -Δu+μ|x|^{-2}u=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm on}\ \ \partialΩ,$$ where $-Δ+\fracμ{|x|^2}$ is the Hardy-Leray operator with $μ\geq -\frac{(N-2)^2}{4}$ and $Ω$ is a smooth bounded domain with $0\inΩ$. We provide lower bounds of $\{λ_{μ,i} \}_i$ together with the Li-Yau's one for $μ>-\frac{(N-2)^2}{4}$ and Karachalio's one for $μ\in [-\frac{(N-2)^2}{4},0)$. Secondly, we obtain Cheng-Yang's type upper bounds for $λ_{μ,k}$. Finally, we get the Weyl's limit of eigenvalues which is independent of the potential's parameter $μ$. 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.