Operator Valued Functions and Boundary Value Problems for the Helmholtz Equation I. Spherical Geometries.

摘要

The boundary integral operator which arises in a double layer formulation of the Neumann problem for the Helmholtz equation is analyzed as an operator valued function of wave number in the particular case of a spherical boundary. The spectrum of the operator is found and its explicit dependence on wave number is exhibited, both analytically and numerically. In addition, the explicit polar decomposition of the operator is carried out and it is shown that asymptotically the operator becomes selfadjoint for small values of wave numbers and unitary for large values. Advantages of operator factorization are discussed. (Author)