Matrix Technique for Nonperturbative Green Function of Time-Independent Schrödinger Equation

Abstract

A Green function of time-independent multi-channel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multi-channel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multi-channel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multi-potential system within quasi-classical approximation. The limits of strong and weak inter-channel interactions are studied. (2000): 15A90, 34B27, 39B42. Mathematics Subject Classifications

Light-induced molecular potentials and temperature-dependent polarization of molecular vapours in intense resonant laser fields: I. Nonperturbative theory in quasi-classical approximation

“…It is important to mention that as the field strength increases, the energies of few levels increase too (for Ne þ 2 these are the levels v d = 6, 9, 12, 13, 17, 18; for Xe þ 2 these are the levels v d = 10,12,15,17,20,23), but the energies of other levels decrease (for Ne þ 2 these are the levels v d = 7, 10, 11, 14, 15, 19, 20; for Xe þ 2 these are the levels v d = 9, 11, 13, 16, 18, 19, 22). There is no regularity in the dependences.…”

“…A general result for a multi-channel Schrö dinger equation [20] demands to search for analytic solutions of Eq. (15) in the form of a 2 · 2 matrix U(R) instead of a vector like Eq.…”