Asymptotic evaluation of the Green's function for large quantum numbers

Abstract

In view of extending the shell model to highly deformed states of heavy nuclei, we discuss the evaluation of the Green's function for the Schrödinger equation in three dimensions with a smooth potential, in the limit of large quantum numbers. Such an evaluation is possible only after smoothing over the energy, which results in the introduction of complex energies. The zeroth order (semiclassical) approximation requires the resolution of the Hamilton-Jacobi equation for complex energies. For a sufficiently large smoothing width, the Green's function is expressed as a local expansion based on the Taylor series expansion of the potential. Higher order corrections to the semiclassical approximation are derived from an integral equation for the Green's function. These corrections produce in particular the reflected waves from the caustic (which generalizes in three dimensions the turning points). The connection with the usual BKW method is discussed in the one-dimensional case by considering the limit of a small smoothing width. The quantum corrections to the Thomas-Fermi model are derived as a first application of these methods.