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: $H_n^{\alpha }=-\Delta +\prod _{i=1}^n|x_i{|}^{{\alpha }_i}$ on ${\mathbb{R}}^n$ ($n>2$), $\alpha :=({\alpha }_1,\cdots ,{\alpha }_n)\in ({\mathbb{R}}_{+}^{\ast }{)}^n$. We apply the results to also derive the leading order spectral asymptotics in the case of the Dirchlet Laplacian $-{\Delta }^D$ on domains ${\Omega }_n^{\alpha }=\{x\in {\mathbb{R}}^n:\prod _{j=1}^n|x_j{|}^{\frac{{\alpha }_j}{{\alpha }_n}}<1\}$. keywords : Trace formulae; Schrödinger operators; Singular asymptotics.

Revised an Published Version