Semiclassical analysis for Hamiltonian in the Born-Oppenheimer approximation

Abstract

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 $% \left\Vert x\right\Vert +\left\Vert y\right\Vert \rightarrow \infty,$ has purely discrete spectrum. We give an application to the harmonic oscillator.