Semiclassical analysis for Hamiltonian in the Born-Oppenheimer approximation

TL;DR

This paper analyzes the spectral properties of a semiclassical Hamiltonian operator within the Born-Oppenheimer approximation, demonstrating it has a purely discrete spectrum under certain conditions and applying the results to the harmonic oscillator.

Contribution

It establishes conditions for the operator to have a purely discrete spectrum and provides an application to the harmonic oscillator.

Findings

Operator has purely discrete spectrum under specified conditions.

Spectral analysis applicable to the harmonic oscillator.

Provides theoretical foundation for semiclassical spectral analysis.

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.