N-body Green's functions and their semiclassical expansion

Abstract

A new semiclassical expansion for quantum mechanics is developed. The high-energy asymptotic expansion for the coordinate-space matrix elements of the $N$-body Green's function is derived. The asymptotic series is characterized by its coefficient functions, $P_n$. It is shown that the coefficient of the $n$th term in the expansion, $P_n$, satisfies a simple recursion relation. The functions, $P_n$, turn out to be polynomials in Planck's constant $h$ of order $2(n-1\mathrm{})\mathrm{}$. In terms of the interaction, the $P_n$ are also polynomials of the potential and its derivatives. If all the $P_n$ are truncated to some common power $M$ in $h$, one generates a natural $M$th order semiclassical approximation to the Green's function. This semiclassical expansion is given a physical interpretation which is particularly simple in terms of state density. By relating the asymptotic series to the Born series, a closed form for the functions $P_n$ is derived.