Quasi-periodic Green’s functions of the Helmholtz and Laplace equations

A classical problem of free-space Green’s function G0Λ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic G0Λ and for the expansion coefficients DL of G0Λ in the basis of regular (cylindrical in two dimensions and spherical in three dimension (3D)) waves, or lattice sums, are reviewed and new results for the case of a one-dimensional (1D) periodicity in 3D are derived. From a mathematical point of view, a derivation of exponentially convergent representations for Schlömilch series of cylindrical and spherical Hankel functions of any integer order is accomplished. Exponentially convergent series for G0Λ and lattice sums DL hold for any value of the Bloch momentum and allow G0Λ to be efficiently evaluated also in the periodicity plane. The quasi-periodic Green’s functions of the Laplace equation are obtained from the corresponding representations of G0Λ of the Helmholtz equation by taking the limit of the wave vector magnitude going to zero. The derivation of relevant results in the case of a 1D periodicity in 3D highlights the common part which is universally applicable to any of remaining quasiperiodic cases. The results obtained can be useful for numerical solution of boundary integral equations for potential flows in fluid mechanics, in many contexts of simulating systems of charged particles, in molecular dynamics, for the description of quasi-periodic arrays of point interactions in quantum mechanics, and in various ab-initio first-principle multiple-scattering theories. PACS numbers: 02.70.Pt; 03.65.Db; 03.65.Nk; 3 4.20.-b; 42.25.Fx; 46.40.Cd; 47.15.km; 61.14.Hg ∗http://www.wave-scattering.com