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The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains

Published online by Cambridge University Press:  24 October 2008

D. S. Jones
D. S. Jones
Affiliation:
Department of MathematicsThe UniversityManchester
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Abstract

The spectrum of −∇2 (and of −∇2 + b) is investigated when the boundary conditions are given on surfaces which extend to infinity. Simple criteria are obtained for determining whether point-eigenvalues are present in the lower part of the spectrum.

Semi-infinite domains which are conical at infinity are found to possess purely continuous spectra when the boundary condition is u = 0 or ∂u/∂v = 0; the radiation condition ensures a unique solution. A counter-example shows that this is not true in general for the boundary condition ∂u/∂v + σu = 0.

Semi-infinite domains which are cylindrical at infinity have a continuous spectrum with a discrete spectrum embedded in it. An example is given.

The results are applied to the theory of surface waves. It is shown that Ursell's ‘trapping modes’ can occur in a canal of finite width when the bed has a protrusion over a finite longth but is otherwise of uniform depth. Trapping modes can also occur when the canal contains a submerged cylinder (not necessarily small in cross-section).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 4 , October 1953 , pp. 668 - 684
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

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