Solution of the Schrödinger equation in terms of classical paths

An expression in terms of classical paths is derived for the Laplace transform $\text{Ω(s)}$ of the Green function G of the Schrödinger equation with respect to $\text{1}\text{h}\text{?}$. For an analytic potential V(r), the function Ω is also analytic in the plane of the complex action variable s; its singularities lie at the values $\text{S}$ of the action along each possible (complex) classical path, including the paths which reflect from singularities of the potential. Accordingly, G may be written as a sum of terms, each of which is associated with such a classical path, and contains the factor $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This expansion formally solves the problem of constructing waves out of the corresponding (complex) classical paths. A similar expression, in terms of closed paths, is derived for the density ? of eigenvalues of the Schrödinger equation. In situations when the eigenvalues are dense, ? is well approximated by the contributions of the shortest closed paths: while the path of vanishing length corresponds to the Thomas-Fermi approximation and its smooth corrections, the other paths yield contributions which oscillate and are damped as $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This result also holds for nonanalytic potentials V(r). If the spectrum is continuous, closed classical paths yield oscillations in the scattering phase-shift. The analysis is also extended to multicomponent wave functions (describing, e.g., motion of particles with spin, or coupled channel scattering); along a classical path, the internal degree of freedom varies adiabatically, except through points at which it is not coupled to the potential, where it may undergo discrete changes.