Path integral representation for inelastic scattering amplitude and its quasiclassical approximations

Different forms of path-integral representation for inelastic, scattering amplitude derived by means of a method suggested earlier by the authors, are discussed. In particular, the path-integral representation for excitation amplitude in the angle-action variables is obtained, as well as the expression for the potential scattering amplitude in the Glauber form. The formulas obtained are convenient for construction of quasiclassical approximations in the framework of the stationary phase method. General quasiclasrsical expression for the excitation amplitude and corresponding eikonal formula including the corrections due to internat motion of the target are given.