NODAL-SURFACE CONJECTURES FOR THE CONVEX QUANTUM BILLIARD

Stemming from known properties of one-dimensional (1-D) and 2-D quantu m billiards, it is conjectured that the nodal surface of the first-exc ited state of the convex 3-D quantum billiard intersects the billiard surface in a single simple closed curve. Examples of the validity of t his conjecture are given for a number of elementary 3-D billiard confi gurations. From these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and c ontain one and only one plane of mirror symmetry normal to the axis of rotation. Two characteristic displacement parameters are defined whic h are labeled an axis length, L, and diameter, a. It is conjectured th at a parameter kappa almost-equal-to 1, exists, whose exact value depe nds on the properties of the billiard, such that for L > kappaa (''pro latelike'') the nodal surface of the first-excited state of a quantum billiard is a plane surface of mirror symmetry which divides the lengt h of the billiard in half. For L < kappaa (''oblatelike'') the nodal s urface of the first-excited state is a plane surface of mirror symmetr y which contains the rotation axis and divides the diameter of the bil liard in half. Arguments are given in support of a third conjecture wh ich addresses the regular polyhedra quantum billiards, termed ''spheri cal-like.'' It is hypothesized that the nodal surface of the first-exc ited state for any of these billiards is any plane of reflection symme try of the given polyhedron.