Semiclassical analysis for Hamiltonian in the Born-Oppenheimer approximation

The purpose of this paper is to show that the operator \begin{equation*} H\left(h\right) =-h^{2}\Delta_{x}-\Delta_{y}+V\left(x,y\right), \end{equation*}% $V$ is continuous (or V\in L^{2}\left(\mathbb{R}_{x}^{n}\times \mathbb{R}%_{y}^{p}\right) ), and $V\left(x,y\right) \rightarrow \infty$ as $$ has purely discrete spectrum. We give an application to the harmonic oscillator.