Four Euclidean conformal group approach to the multiphoton processes in the H-atom

Abstract

A compact analytical form, suitable for any analytic continuation, is obtained for the following bound–bound N‐photon transition matrix element, Inlm→n′l′m′ ≡〈n′l′m′ ‖ ∏i = 1N {p⋅eir⋅ei}eiki⋅ r ×G(Ei)‖nlm〉, where G(E) is the Coulomb Green’s function. We show that Inlm→n′l′m′ is a ’’linear superposition’’ of matrix elements T1(g)n′l′m′→nlm of some irreducible representation T1 of a semigroup G−1? contained in the four Euclidean conformal group G = SU*(4)≊SO0(1,5). This ’’linear superposition’’ is understood in the general framework of the theory of the distributions on a Lie group. The final result is a linear combination of special functions known as ’’generalized Euler functions.’’