Higher dimensional non standard eigenvalue asymptotics

Abstract

In this article we extend B. Simon's construction and results for leading order eigenvalue asymptotics to $n$-dimensional Schrödinger operators with non-confining potentials given by: $H^α_n=-Δ+\prod\limits_{i=1}^n |x_i|^{α_i}$ on $\mathbb{R}^n$ ($n>2$), $α:=(α_1,\cdots,α_n)\in (\mathbb{R}_{+}^*)^n$. We apply the results to also derive the leading order spectral asymptotics in the case of the Dirchlet Laplacian $-Δ^D$ on domains $Ω^α_n=\{x\in\mathbb{R}^n: \prod\limits_{j=1}^n |x_j|^{\frac{α_j}{α_n}}<1 \}$. keywords : Trace formulae; Schrödinger operators; Singular asymptotics.