WKB approximation and threshold law for electron-atom ionization

The solutions of the Hamilton-Jacobi equation and of the continuity equation for electron-atom ionization problem at zero total angular momentum are investigated in the neighbourhood of saddle points of the potential. Solutions are found which describe sets of orbits whose points of condensation coincide with the saddle points of the potential. The asymptotic behaviour of solutions corresponds to a spherically outgoing wave or to a plane wave. The energy dependence of WKB wave functions is determined and by means of a matching procedure the Wannier ionization threshold law has been confirmed. The connection between the WKB approximation and the asymptotic expansion of the zero- energy wave function has been considered.