SPECTRAL THEORY FOR THE WAVE-EQUATION IN 2 ADJACENT WEDGES

Consider the two adjacent rectangular wedges K-1, K-2 with common edge in the upper halfspace of R-3 and the operator A (=-Laplacian multipl ied by different constant coefficients a(1), a(2) in K-1, K-2, respect ively) acting on a subspace of (j=1)Pi(2) L-2(K-j). This subspace shou ld consist of those sufficiently regular functions u=(u(1),u(2)) satis fying the homogeneous Dirichlet boundary condition on the bottom of th e upper halfspace. Moreover, the coincidence of u, and u, along the in terface of the two wedges is prescribed as well as a transmission cond ition relating their first one-sided derivatives. We interpret the cor responding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different materi al constants. In this paper it is shown (by Friedrichs' extension) tha t A is selfadjoint in a suitable Hilbert space. Applying the Fourier ( -sine) transformations we reduce our problem with singularities along the z-axis to a non-singular Klein-Gordon equation in one space dimens ion with potential step. The resolvent, the limiting absorption princi ple and expansion in generalized eigenfunctions of A are derived (by P lancherel theory) from the corresponding results concerning the latter equation in one space dimension. An application of the spectral theor em for unbounded selfadjoint operators on Hilbert spaces yields the so lution of the time dependent problem with prescribed initial data. The paper is concluded by a discussion of the relation between the physic al geometry of the problem and its spectral properties. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.