Bounds of Dirichlet eigenvalues for Hardy-Leray operator

Abstract

The purpose of this paper is to study the eigenvalues $\{{\lambda }_{\mu ,i}{\}}_i$ for the Dirichlet Hardy-Leray operator, i.e. $$-\Delta u+\mu |x{|}^{-2}u=\lambda u\mathrm{in}\Omega ,u=0\mathrm{on}\partial \Omega ,$$ where $-\Delta +\frac{\mu }{|x{|}^2}$ is the Hardy-Leray operator with $\mu \ge -\frac{(N-2{)}^2}{4}$ 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.

20 pages