Behavior of eigenvalues of certain Schrödinger operators in the rational Dunkl setting

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

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