Semiclassical analysis for Hamiltonian in the Born-Oppenheimer approximation

Abstract

The purpose of this paper is to show that the operator H (h) = −h^2 × (delta)x − (delta)y + V (x, y) ,V is continuous (or V such that L^2 `Rnx × Rpy´), and V (x, y) approches infinity as ||x|| + ||y||approches infinity, has purely discrete spectrum. We give an application to the harmonic oscillator. Keywords: Discrete spectrum; harmonic oscillator; locally compact operator. 2010 Mathematics Subject Classification: 35J10, 35Q55