High order corrections to the time-dependent Born-Oppenheimer approximation. I. Smooth potentials


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 $\varepsilon^{-4}$, then certain solutions to the time dependent Schrödinger equation have asymptotic expansions to arbitrarily high order in powers of $\varepsilon$, and as $\varepsilon \searrow 0$. The zero-th order terms in these equations are the wave functions of the usual time-dependent Born-Oppenheimer approximation.


George A. Hagedorn