We derive analytical expressions for the Weyl formulas of two-and three-dimensional raysplitting billiards. Our analytical results are in excellent agreement with numerical checks.
1998 Academic Press
I. INTRODUCTION
The knowledge of the asymptotic behavior of the number of modes in a resonator is of fundamental importance in many fields of physics . It is needed, e.g., for the derivation of Planck's radiation law, the evaluation of the Casimir force, or the computation of the specific heat of solids. The origin of the problem dates back to 1910 when H. A. Lorentz, in a talk delivered at Go ttingen, conjectured that to leading order the number of radiation modes in a resonator below a frequency f depends only on the volume of the resonator and the cube of f. The mathematician D. Hilbert, who was present in the audience commented that in his opinion a mathematical proof of this conjecture is so difficult that it would surely not be accomplished during his lifetime . This was a dangerous statement since not a year later H. Weyl managed to prove Lorentz's conjecture . Following this proof, Weyl managed to refine his theorem in a number of subsequent papers [4 8]. Weyl's seminal work is nowadays recognized to be of fundamental importance in mathematics and physics and has been extended by many authors (see, e.g., Refs. ).
In this paper we present a collection of Weyl formulas for quantum ray-splitting billiards. Ray splitting is a relatively recent direction in quantum chaos research that started with the seminal paper by Couchman et al. [10]. Ray splitting is a universal feature of all wave systems with discontinuities in the propagation medium or the system potentials. As a basis for the quantum mechanics and the quantum chaology of ray-splitting systems knowledge of their Weyl formulas is indispensible. Thus, connecting with the work of Weyl, we are interested in the asymptotic behavior of the spectra of quantum ray-splitting billiards, i.e., the quantum spectra of bounded domains with potential discontinuities in their interior.