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Decaying modes for the wave equation in the exterior of an obstacle
Author(s) -
Lax P. D.,
Phillips R. S.
Publication year - 1969
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160220603
Subject(s) - mathematics , eigenvalues and eigenvectors , mathematical analysis , dirichlet eigenvalue , upper and lower bounds , dirichlet distribution , wave equation , boundary (topology) , obstacle , monotonic function , obstacle problem , dirichlet boundary condition , boundary value problem , dirichlet's principle , quantum mechanics , physics , political science , law
In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ k } of Δ on the geometry of the obstacle . In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle , both for the Dirichlet and Neumann boundary conditions . From this we deduce, by comparison with spheres—for which the eigenvalues {λ k } can be determined as roots of special functions—upper and lower bounds for the density of the real {λ k }, and upper and lower bounds for λ 1 , the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on . Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono‐energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.