Dunkl-Schr\"odinger operators

In this paper, we consider the Schr"odinger operators $L_k=-{\mathrm{\Delta }}_k+V$, where ${\mathrm{\Delta }}_k$ is the Dunkl-Laplace operator and $V$ is a non-negative potential on $R^d$. We establish that $L_k$ is essentially self-adjoint on $C_0^{\mathrm{\infty }}$. In particular, we develop a bounded $H^{\mathrm{\infty }}$-calculus on $L^p$ spaces for the Dunkl harmonic oscillator operator.