The Method of Resolvent Equations

Abstract

Generally speaking, the perturbation theory which forms the heart of theoretical methods for treating off-resonant phenomena cannot be applied directly in the case of resonant multiphoton processes. Modifications are called for to correctly and conveniently approach the problem of resonance breakdown of ordinary perturbation theory. This is most easily carried out by adopting^(108–112) the powerful method of resolvent equations^{(1,10,113,114)} to the multiphoton transition problem. In this chapter a system of algebraic equations satisfied by the matrix elements of the resolvent for the semiclassical as well as for the number-state description of the multiphoton Hamiltonian is developed. Both monomode and multimode resolvent equations are constructed. A rather general model-Hamiltonian for multiphoton processes is introduced, and the resulting resolvent equations are solved explicitly for the four kinds of multiphoton transition of physical interest. Probability distributions of occupation of the above-threshold continuum states by multiphoton absorption from a bound state are also given.