Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity

MV Berry, MR Dennis

Research output: Contribution to journalArticle (Academic Journal)peer-review

30 Citations (Scopus)


Waves 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.
Translated title of the contributionBoundary-condition-varying circle billiards and gratings: the Dirichlet singularity
Original languageEnglish
Pages (from-to)135203-1 - 135203-23
Number of pages23
JournalJournal of Physics A: Mathematical and Theoretical
Publication statusPublished - Apr 2008

Bibliographical note

Publisher: Institute of Physics Publishing


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