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 contribution||Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity|
|Pages (from-to)||135203-1 - 135203-23|
|Number of pages||23|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Publication status||Published - Apr 2008|