Generalized Sturm expansions of the Coulomb Green's function and two-photon Gordon formulas

The radial component of the Coulomb Green's function (CGF) is written in th e form of a double series in Laguerre polynomials (Sturm's functions in the Coulomb problem), which contains two free parameters alpha and alpha'. The obtained result is applicable both in the nonrelativistic case and for the CGF of the squared Dirac equation with a Coulomb potential. The CGF is dec em posed into the resonance and potential components (the latter is a smoot h function of energy) for alpha = alpha'. In the momentum representation, t he CGF with the free parameters is written in the form of an expansion in f our-dimensional spherical functions. The choice of the parameters alpha and alpha' in accordance with the specific features of the given problem radic ally simplifies the calculation of the composite matrix elements for electr omagnetic transitions. Closed analytic expressions (in terms of hypergeomet ric functions) are obtained for the amplitudes of bound-bound and bound-fre e two-photon transitions in the hydrogen atom from an arbitrary initial sta te \nl], which generalize the known (one-photon) Gordon formulas. The dynam ic polarizability tensor components alpha (n/m)(omega) for an arbitrary n a re expressed in terms of the hypergeometric function F-2(1) depending only on iota and <(<omega>)over bar> and through the polynomial functions f(nl)( <(<omega>)over tilde>) of frequency <(<omega>)over tilde> = (h) over bar om ega/\E-n\. The Rydberg (n much greater than 1) and threshold ((h) over bar omega similar to \E-n\) asymptotic forms of polarizabilities are investigat ed. (C) 2001 MAIK "Nauka/Interperiodica".