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 \(dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}\,d{\mathbf {x}}\) the associated measure in \({\mathbb {R}}^N\). Let \(L=-\varDelta +V\), \(V\ge 0\), 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.