Circular-sector quantum-billiard and allied configurations

The circular‐sector quantum‐billiard problem is studied. Numerical evaluation of the zeros of first‐order Bessel functions finds that there is an abrupt change in the nodal‐line structure of the first excited state of the system (equivalently, second eigenstate of the Laplacian) at the critical sector‐angle θc=0.354π. For sector‐angle θ0, in the domain 0<θ0<θc, the nodal curve of the first excited state is a circular‐arc segment. For θc<θ0≤π, the nodal curve of the first excited state is the bisector of the sector. Otherwise nondegenerate first excited states become twofold degenerate at the critical‐angle θc. The ground‐ and first‐excited‐state energies (EG,E1) increase monotonically as θ0 decreases from its maximum value, π. A graph of E1 vs θ0 reveals an inflection point at θ0=θc, 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 functions whose respective orders differ by a noninteger. Applicat...