M V Berry and M R Dennis 2008 J. Phys. A: Math. Theor. 41 135203 doi:10.1088/1751-8113/41/13/135203
Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity
M V Berry and M R Dennis
Show affiliationsWaves in a two-dimensional domain with Robin (mixed) boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena. If the variation includes a 'D point' where the boundary condition is Dirichlet (vanishing wavefunction), a variety of arguments indicate that the system is singular. For a circle billiard, the boundary condition fails to determine a discrete set of levels, so the spectrum is continuous. For a diffraction grating defined by periodically varying boundary conditions on the edge of a half-plane, the phase of a diffracted beam amplitude remains undetermined. In both cases, the wavefunction on the boundary has a singularity at a D point, described by the polylogarithm function.
- PACS
- MSC
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58Kxx Theory of singularities and catastrophe theory (See also 32Sxx, 37-XX)
78A45 Diffraction, scattering (See also 34E20 for WKB methods)
37D50 Hyperbolic systems with singularities (billiards, etc.)
- Subjects
- Dates
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Issue 13 (4 April 2008)
Received 21 December 2007
Published 14 March 2008
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Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity
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