Exponentially accurate error estimates of quasiclassical eigenvalues. II. Several dimensions

Abstract

We study the behavior of truncated Rayleigh-Schrodinger series for low-lying eigenvalues of the time-independent Schrodinger equation, in the semiclassical limit (h) over bar SE arrow0. In particular we prove that if the potential energy satisfies certain conditions, there is an optimal truncation of the series for the eigenvalues, in the sense that this truncation is exponentially close to the exact eigenvalue. These results were already discussed for the one-dimensional case in a previous article. This time we consider the multi-dimensional problem, where degeneracy plays a central role. (C) 2003 American Institute of Physics.