## High order corrections to the time-dependent Born-Oppenheimer approximation. I: Smooth potentials.(English)Zbl 0619.35094

Ann. Math. (2) 124, 571-590 (1986); erratum ibid. 126, 219 (1987).
We consider the dynamics of a quantum mechanical system which consists of some particles of large mass and some particles of small mass, which interact through smooth potentials. We prove that if the large masses are proportional to $$\epsilon^{-4}$$, then certain solutions to the time dependent Schrödinger equation have asymptotic expansions to arbitrarily high order in powers of $$\epsilon$$, and as $$\epsilon\searrow 0$$. The zeroth order terms in these expansions are the wave functions of the usual time-dependent Born-Oppenheimer approximation.

### MSC:

 35Q99 Partial differential equations of mathematical physics and other areas of application 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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