Bounds of Dirichlet eigenvalues for Hardy-Leray operator
Huyuan Chen, Feng Zhou · Discrete and Continuous Dynamical Systems - S · 2023
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 $, as well as the Li-Yau's one when $ \mu>-\frac{(N-2)^2}{4} $ and Karachalios's bounds for $ \mu\in [-\frac{(N-2)^2}{4},0) $. Secondly, we obtain Cheng-Yang's type upper bounds for $ \lambda_{\mu,k} $. Additionally, we get the Weyl's limit of eigenvalues which is independent of the potential's parameter $ \mu $. This interesting phenomenon indicates that the inverse-square potential does not play an essential role for the asymptotic behavior of the spectrum of the problem under study.