Semiclassical Analysis for Hamiltonianin the Born-Oppenheimer Approximation

The purpose of this paper is to show that the operator H (h) = −h^2Δx −Δy+V (x, y), V is continuous (or V (element of) L^2 (Rnx × Rpy), and V (x, y)→(infinity)as llxll+llyll→(infinity) has purely discrete spectrum. We give an application to the harmonic oscillator.