Symmetry of emission spectra of dual convex polyhedra

Abstract

We consider convex polyhedra which may be symmetrically inscribed in a sphere (‘T-polyhedra’). Incorporating elements of group theory, one of Courant's theorems and Weyl's expansion for the density of states, it is argued that a T-polyhedron quantum billiard and its dual of equal volume have equal emission frequencies. Application is made to the infinite set of convex ‘equilateral zonohedra’ quantum billiards. A thin-film resonant absorption experiment is proposed to verify these findings. It is found that, for dual convex regular polygons in , wave functions of the dual polygon in the related billiard problem maintain their functional form and, for duals of larger area, eigenvalues are lowered by a common scale factor.

Acknowledgements

Fruitful discussions on these topics with Mason Proter, Jack Freed, Robert Connelly, Gregory Ezra, Vaclav Kostroun and Veit Elser are gratefully acknowledged.