A succinct way to diagonalize the translation matrix in three dimensions

The fast multipole method is an effective way to expedite iterative solutions of integral equations for electrodynamic and elastodynamic problems. Iterative solvers, such as the conjugate gradient method, require matrix vector multiplies, and for dense matrices, these matrix-vector multiplies constitute the predominant computational cost as well as requiring a large memory. The fast multipole method is based on the translation of multipoles from one coordinate system to another, which is achieved by using translation matrices. However, the mere use of translation matrices does not reduce the computational cost nor the memory requirement for dynamic problems involving surface scatterers. For such problems, the crucial step in the fast multipole method is the diagonalization of the translation operators. The translation matrix for the three dimensional Helmholtz wave equation has been successfully diagonalized using an alternative and succinct method. The method reveals the relationship between the translation matrices and their representation of the translation group. A diagonalization is expected under a plane-wave basis for the representation since a plane-wave basis forms an irreducible representation for the translation group. Hence, the diagonalization of the translation matrices from the spherical harmonic representation can be viewed as a series of similarity transforms. The result can be used in the fast multipole method and the multilevel fast multipole method where multiple scattering involves interaction between multipoles.