The classical mechanics, exact quantum mechanics and semiclassical quantum
mechanics of the billiard in the triaxial ellipsoid are investigated. The s
ystem is separable in ellipsoidal coordinates. A smooth description of the
motion is given in terms of a geodesic now on a solid torus, which is a fou
rfold cover of the interior of the ellipsoid. Two crossing separatrices lea
d to four generic types of motion. The action variables of the system are i
ntegrals of a single Abelian differential of second kind on a hyperelliptic
curve of genus 2. The classical separability carries over to quantum mecha
nics giving two versions of generalized Lame equations according to the two
sets of classical coordinates. The quantum eigenvalues define a lattice wh
en transformed to classical action space. Away from the separatrix surfaces
the lattice is given by EBK quantization rules for the four types of class
ical motion. The transition between the four lattices is described by a uni
form semiclassical quantization scheme based on a WKB ansatz. The tunneling
between tori is given by penetration integrals which again are integrals o
f the same Abelian differential that gives the classical action variables.
It turns out that the quantum mechanics of ellipsoidal billiards is semicla
ssically most elegantly explained by the investigation of its hyperelliptic
curve and the real and purely imaginary periods of a single Abelian differ
ential. (C) 1999 Academic Press.