Spectral theory for the wave equation in two adjacent wedges
✍ Scribed by Felix Ali Mehmeti; Erhard Meister; Krešo Mihalinčić
- Publisher
- John Wiley and Sons
- Year
- 1997
- Tongue
- English
- Weight
- 238 KB
- Volume
- 20
- Category
- Article
- ISSN
- 0170-4214
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✦ Synopsis
Consider the two adjacent rectangular wedges K , K with common edge in the upper halfspace of 1 and the operator A ("!Laplacian multiplied by different constant coefficients a , a in K , K , respectively) acting on a subspace of H ¸(K H ). This subspace should consist of those sufficiently regular functions u"(u , u ) satisfying the homogeneous Dirichlet boundary condition on the bottom of the upper halfspace. Moreover, the coincidence of u and u along the interface of the two wedges is prescribed as well as a transmission condition relating their first one-sided derivatives. We interpret the corresponding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different material constants. In this paper it is shown (by Friedrichs' extension) that 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 dimension with potential step. The resolvent, the limiting absorption principle and expansion in generalized eigenfunctions of A are derived (by Plancherel theory) from the corresponding results concerning the latter equation in one space dimension. An application of the spectral theorem for unbounded selfadjoint operators on Hilbert spaces yields the solution of the time dependent problem with prescribed initial data. The paper is concluded by a discussion of the relation between the physical geometry of the problem and its spectral properties.