Distribution of eigenfrequencies for the wave equation in a finite domain. 1. Three-dimensional problem with smooth boundary surface

Mar, 1970
47 pages
Published in:
  • Annals Phys. 60 (1970) 401-447

Citations per year

197219851998201120240246810
Abstract: (Elsevier)
The distribution of eigenvalues of the equation Δϕ + Eϕ = 0 is calculated for a volume V of arbitrary shape, and for the general boundary condition ∂ϕ ∂n = κϕ on the surface S , assumed to be smooth. A time-independent Green function method is used, involving a multiple reflection expansion. In the limit of wavelengths small compared to any characteristic dimension of the system, the eigenvalue density, smoothed to eliminate its fluctuating part, is given by the asymptotic expansion: ϱ (E) = 1 4π 2 Vk+S π 4 −δ + 1 k 1 3 + cos 2 δ−δ cot δ ∫dσ 1 2 1 R 1 + 1 R 2 +… where R 1 and R 2 are the main curvature radii of S , with δ = tan −1 κ k . Both surface and curvature terms depend on the boundary condition. The correct curvature term differs from the extrapolation of the wedge term of the parallelepiped, which has sometimes been used in nuclear physics. Neumann and Dirichlet boundary conditions are recovered for κ = 0 and κ → + ∞. For κ → − ∞, the density contains an additional contribution corresponding to surface states.