Generalized Orthogonality Relation for Spherical Harmonics

One of us (G.G), with collaborators, has been involved in the study of generalized Lorenz-Mie theories (GLMTs) describing the interaction between electromagnetic arbitrary shaped beams (typically laser beams) and a class of regular scatterers for which solutions to Maxwell’s equations can be found by using the method of separation of variables, e.g. Gouesbet & Grehan (2000a), J.A.Lock & Gouesbet (2009), Gouesbet (2009a), and references therein. It has, at a certain time, been found interesting to examine whether the knowledge gained in the effort of developing electromagnetic GLMTs could be, at least partially, adapted to quantum mechanical problems. The examination of the issue of quantum scattering of quantum arbitrary shaped beams produced several papers devoted to (i) the description of quantum arbitrary shaped beams Gouesbet (2005), Gouesbet & J.A.Lock (2007) (ii) the evaluation of cross-sections in the case of quantum arbitrary shaped beams interacting with radial quantum potentials Gouesbet (2006a), Gouesbet (2007a) and (iii) the exhibition of formal cross-sectional analogies between electromagnetic and quantum scatterings Gouesbet (2004), Gouesbet (2006b), Gouesbet (2007b). During the development of this work, the evaluation of two integrals, based on spherical harmonics, has been required. These integrals may be obtained from a generalized orthogonality relation for spherical harmonics which is established in this paper. Another recent application concerns the optical theorem and non plane wave scattering in quantum mechanics Gouesbet (2009b).