  In this paper, we consider the Schr\"odinger operators $L_k=-\Delta_k+V$,
where $\Delta_k$ 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
$C_0^\infty$. In particular, we develop a bounded $H^\infty$-calculus on $L^p$
spaces for the Dunkl harmonic oscillator operator.
