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 $-{\mathrm{\Delta }}^D$ 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.