Basis Properties of Traces and Normal Derivatives of Spherical-Separable Solutions of the Helmholtz Equation

Dallas, A.G.
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Department of Mathematical Sciences
The classical solutions of the Heimholtz equation resulting from the separation-of-variables procedure in spherical coördinates are frequently used in one way or another to approximate other solutions. In particular, traces and/or normal derivatives of certain sequences of these spherical-separable solutions are commonly used as trial-and-test-functions in Galerkin procedures for the approximate solution of boundary-operator problems arising from the reformulation of exterior or interior boundary-value problems and set on the boundary Γ of the domain where a solution is wanted. While the completeness properties of these traces and normal derivatives in the usual Hilbert space L2( Γ) are well known, their basis properties are not. We show that such sequences of traces or normal derivatives of the outgoing spherical-separable solutions form bases for L2( Γ) only when Γ is a sphere centered at the pole of the spherical solutions; corresponding results are given for the entire solutions, accounting for the possibility of an interior eigenvalue. We identify other Hilbert spaces, connected with the far-field pattern, for which these functions do provide bases. We apply the results to discuss some aspects of the Waterman schemes for approximate solutions of scattering problems (the so-called “T-matrix method”), including the previous article of KRISTENSSON, RAMM, and STRÖM (J. Math. Phys.24 (1983), 2619-2631) on the convergence of such methods.