Feynman path integrals and the trace formula for the Schroedinger operators

We study Schroedinger operators of the form H = h2/2Δ + 1/2x x A2x + V(x) on Rsup(d), where A2 is a strictly positive symmetric d x d matrix and V(x) is a continuous real function which is the Fourier transform of a bounded measure. If lambdasub(n) are the eigenvalues of H we show that the theta function THETA(t) = Σsub(n)exp(-i/htlambdasub(n)) is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integral over the Hilbert space of closed trajectories). We use these explicit expressions to give the asymptotic behaviour of THETA(t) for small h in terms of classical periodic orbits, thus obtaining a trace formulae for the Schroedinger operators. This then yields an asymptotic expansion of the spectrum of H in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds. (orig.)