CIRCULAR-SECTOR QUANTUM-BILLIARD AND ALLIED CONFIGURATIONS

The circular-sector quantum-billiard problem is studied. Numerical eva luation Of the zeros of first-order Bessel functions finds that there is an abrupt change in the nodal-line structure of the first excited s tate of the system (equivalently, second eigenstate of the Laplacian) at the critical sector-angle theta(c) = 0.354 pi. For sector-angle the ta0, in the domain 0 < theta0 < theta(c), the nodal curve of the first excited state is a circular-arc segment. For theta(c) < theta0 less-t han-or-equal-to pi, the nodal curve of the first excited state is the bisector of the sector. Otherwise nondegenerate first excited states b ecome twofold degenerate at the critical-angle theta(c). The ground- a nd first-excited-state energies (E(G), E1) increase monotonically as t heta0 decreases from its maximum value, pi. A graph of E1 vs theta0 re veals an inflection point at theta0 = theta(c), which is attributed to the change in Bessel-function contribution to the development of E1. A proof is given for the existence of a common zero for two Bessel fun ctions whose respective orders differ by a noninteger. Application of these results is made to a number of closely allied quantum-billiard c onfigurations.