Behavior of eigenvalues of certain Schr\"odinger operators in the rational Dunkl setting

Abstract

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. We denote by $dw(\mathbf{x})=\Pi_{\alpha \in R}|\langle \mathbf{x},\alpha \rangle|^{k(\alpha)}\,d\mathbf{x}$ the associated measure in $\mathbb{R}^N$. Let $L=-\Delta +V$, $V\geq 0$, be the Dunkl--Schr\"odinger operator on $\mathbb R^N$. Assume that there exists $q >\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse H\"older class ${\rm{RH}}^{q}(dw)$. For $\lambda>0$ we provide upper and lower estimates for the number of eigenvalues of $L$ which are less or equal to $\lambda$. Our main tool in the Fefferman--Phong type inequality in the rational Dunkl setting.