Exponentially Small Error Estimates of Quasiclassical Eigenvalues

Julio Toloza Virginia Tech

We study the behavior of truncated Rayleigh-Schrödinger series for the low-lying eigenvalues of the time-independent Schrödinger equation, in the semiclassical limit ℏ ↘ 0. We prove that, under certain hypotheses on the potential V (x) and for small ℏ > 0, the perturbation series for the eigenvalues admits an exponentially accurate truncation. That is, the difference between the truncated series and the actual eigenvalue can be made smaller than exp(−C/ℏ) for a positive constant C. An analogous statement is shown for the corresponding eigenfuctions.