PERIODIC ORBIT THEORY ANALYSIS OF A CONTINUOUS FAMILY OF QUASI-CIRCULAR BILLIARDS

We compute the Fourier transform (rho(L)) of the quantum mechanical en ergy level density for the problem of a particle in a two-dimensional circular infinite well (or circular billiard) as well as for several s pecial generalizations of that geometry, namely the half-well, quarter -well, and the circular well with a thin, infinite wall along the posi tive x-axis (hereafter called a circular well plus baffle). The result ing peaks in plots of \rho(L)\(2) versus L are compared to the lengths of the classical closed trajectories in these geometries as a simple example of the application of periodic orbit (PO) theory to a billiard or infinite well system. We then solve the Schrodinger equation for t he general case of a circular well with infinite walls both along the positive x-axis and at an arbitrary angle Theta (a circular ''slice'') for which the half-well (Theta = pi), quarter-well (Theta = pi/2), an d circular well plus baffle (Theta = 2 pi) are then all special cases. We perform a PO theory analysis of this general system and calculate \rho(L)\(2) for many intermediate values of Theta to examine how the p eaks in rho(L) attributed to periodic orbits change as the quasi-circu lar wells are continuously transformed into each other. We explicitly examine the transitions from the half-circular well to the circle plus baffle case (half-well to quarter-circle case) as Theta changes conti nuously from pi to 2 pi (from pi to pi/2) in detail. We then discuss t he general Theta --> 0 limit, paying special attention to the cases wh ere Theta = pi/2(n), as well as deriving the formulae for the lengths of closed orbits for the general case. We find that such a periodic or bit theory analysis is of great benefit in understanding and visualizi ng the increasingly complex pattern of closed orbits as Theta --> 0. ( C) 1998 American Institute of Physics.