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

Abstract

For a normalized root system $R$ in ${\mathbb{R}}^N$ and a multiplicity function $k\ge 0$ let $\mathbf{N}=N+{\sum }_{\alpha \in R}k(\alpha )$. We denote by $dw(\mathbf{x})={\Pi }_{\alpha \in R}|⟨\mathbf{x},\alpha ⟩{|}^{k(\alpha )}d\mathbf{x}$ the associated measure in ${\mathbb{R}}^N$. Let $L=-\Delta +V$, $V\ge 0$, 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 ${\mathrm{RH}}^q(dw)$. 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.

arXiv comment12 pages. arXiv admin note: text overlap with arXiv:1912.11352, arXiv:1910.06433