The Laplace-Beltrami operator on surfaces with axial symmetry

The physical situation which has initiated this research is that of a dielectric particle with electric charges on its surface, placed in electric field. Here, the diffusion equation of the charges is coupled with the Maxwell equations. There is an analytical solution of this system of equation [1] which involves some functional calculus with operators, in particular with Laplace-Beltrami operator defined on the surface of that particle. We can imagine many other physical situations described by a complicated system of equations where the Laplace-Beltrami operator is implicated (e.g. that of the acoustic wave scattering on particles with membrane, etc.). As before, one can find a compact solution by using functional calculus. However, these solutions are not complete because, at this level, all is formal. We must have an effective procedure to calculate the expressions which involve operators. One can try