On the Completeness of Sets of Solutions to the Helmholtz Equation

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Abstract
Completeness in L2(δD) is established for sets of functions formed from solutions to the two-dimensional Helmholtz equation in a domain D. Each function is a linear combination of a solution (found by separation of variables) and its normal derivative on δD, so the sets may be used to solve impedance-type boundary value problems. Sets that contain either regular Bessel functions or singular Hankel functions are considered. Methods of proof are employed that provide alternatives to the conventional potential-theoretic approaches. In the majority of cases, the domain of interest is bounded and simply connected. One completeness result for a bounded, doubly-connected domain is proved. In some circumstances, one of the methods leads to a mild but inessential eigenvalue restriction.

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