On \({\mathbb {R}}^N\) equipped with a normalized root system R, a multiplicity function \(k(\alpha ) > 0\) , and the associated measure \(\begin{aligned} dw({\textbf{x}})=\prod _{\alpha \in R}|\langle {\textbf{x}},\alpha \rangle |^{k(\alpha )}\, d{\textbf{x}}, \end{aligned}\) we consider a Dunkl Schrödinger operator \(L=-\Delta _k+V\) , where \(\Delta _k\) is the Dunkl Laplace operator and \(V\in L^1_{\textrm{loc}} (dw)\) is a non-negative potential. Let \(h_t({\textbf{x}},{\textbf{y}})\) and \(k^{\{V\}}_t({\textbf{x}},{\textbf{y}})\) denote the Dunkl heat kernel and the integral kernel of the semigroup generated by \(-L\) respectively. We prove that \(k^{\{V\}}_t({\textbf{x}},{\textbf{y}})\) satisfies the following heat kernel lower bounds: there are constants \(C, c>0\) such that \(\begin{aligned} h_{ct}({\textbf{x}},{\textbf{y}})\le C k^{\{V\}}_t({\textbf{x}},{\textbf{y}}) \end{aligned}\) if and only if \(\begin{aligned} \sup _{{\textbf{x}}\in {\mathbb {R}}^N} \int _0^\infty \int _{{\mathbb {R}}^N} V({\textbf{y}})w(B({\textbf{y}},\sqrt{t}))^{-1} e^{-\Vert {\textbf{x}}-{\textbf{y}}\Vert ^2/t}\, dw({\textbf{y}})\, dt<\infty , \end{aligned}\) where \(B({ {\textbf{x}}},\sqrt{t})\) stands for the Euclidean ball centered at \({\textbf{x}} \in \mathbb {R}^N\) and radius \(\sqrt{t}\) .
