BOUNDS OF DIRICHLET EIGENVALUES FOR HARDY-LERAY OPERATOR

The purpose of this paper is to study the eigenvalues {λµ,i}i for the Dirichlet Hardy-Leray operator, i.e. −∆u + µ|x|⁻²u = λu in Ω, u = 0 on ∂Ω, µ where −∆ + _|x_|2 is the Hardy-Leray operator with µ ≥ − ^(N−₄²⁾² and Ω is a smooth bounded domain with 0 ∈ Ω. We provide lower bounds of {λµ,i}i, as well as the Li-Yau’s one when µ > − ^(N−₄²⁾² and Karachalios’s bounds for µ ∈ [− ^(N−₄²⁾² , 0). Secondly, we obtain Cheng-Yang’s type upper bounds for λ_µ,k. Additionally, we get the Weyl’s limit of eigenvalues which is independent of the potential’s parameter µ. 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.