Higher dimensional non standard eigenvalue asymptotics

In this article we extend B. Simon's construction and results for leading order eigenvalue asymptotics to $n$-dimensional Schr\"odinger operators with non-confining potentials given by: $H^\alpha_n=-\Delta +\prod\limits_{i=1}^n |x_i|^{\alpha_i}$ on $\mathbb{R}^n$ ($n>2$), $\alpha:=(\alpha_1,\cdots,\alpha_n)\in (\mathbb{R}_{+}^*)^n$. We apply the results to also derive the leading order spectral asymptotics in the case of the Dirchlet Laplacian $-\Delta^D$ on domains $\Omega^\alpha_n=\{x\in\mathbb{R}^n: \prod\limits_{j=1}^n |x_j|^{\frac{\alpha_j}{\alpha_n}}<1 \}$. keywords : Trace formulae; Schr\"odinger operators; Singular asymptotics.