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We study the behaviour of truncated Rayleigh-Schrödinger series for the low-lying eigenvalues of the one-dimensional, time-independent Schrödinger equation, in the semiclassical limith → 0. Under certain hypotheses on the potential V (x), we prove that for any given smallh > 0 there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and exact eigenvalue is smaller than exp(−C/h) for some positive constant C. We also prove the analogous results concerning the eigenfunctions.

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