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.

对于归一化的根系统ř在\（{\ mathbb {R}} ^ N \）和多个功能\（K \ GE 0 \）让\（{\ mathbf {N}} = N + \总和_ {\阿尔法\in R} k(\alpha )\)。我们表示为\(dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )} \,d{\mathbf {x}}\)在\({\mathbb {R}}^N\) 中的相关度量。设\(L=-\varDelta +V\) , \(V\ge 0\)是\({\mathbb {R}}^N\)上的 Dunkl–Schrödinger 算子。假设存在\(q >\max (1,\frac{{\mathbf {N}}}{2})\)使得V属于反向 Hölder 类\({\mathrm{RH}}^{q}(dw)\)。对于\(\lambda >0\)，我们提供L的特征值数量的上限和下限估计值，这些值小于或等于\(\lambda \)。我们在有理 Dunkl 设置中 Fefferman-Phong 型不等式的主要工具。