Periodic orbit theory analysis of a continuous family of quasi-circular billiards Available to Purchase

We compute the Fourier transform $(ρ(L))$ of the quantum mechanical energy level density for the problem of a particle in a two-dimensional circular infinite well (or circular billiard) as well as for several special generalizations of that geometry, namely the half-well, quarter-well, and the circular well with a thin, infinite wall along the positive $x$-axis (hereafter called a circular well plus baffle). The resulting peaks in plots of $|ρ(L)|2$ 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 Schrödinger equation for the general case of a circular well with infinite walls both along the positive $x$-axis and at an arbitrary angle Θ (a circular “slice”) for which the half-well (Θ=π), quarter-well (Θ=π/2), and circular well plus baffle $(Θ=2π)$ are then all special cases. We perform a PO theory analysis of this general system and calculate $|ρ(L)|2$ for many intermediate values of Θ to examine how the peaks in $ρ(L)$ attributed to periodic orbits change as the quasi-circular 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 Θ changes continuously from π to 2π (from π to π/2) in detail. We then discuss the general Θ→0 limit, paying special attention to the cases where $Θ=π/2n$⁠, as well as deriving the formulae for the lengths of closed orbits for the general case. We find that such a periodic orbit theory analysis is of great benefit in understanding and visualizing the increasingly complex pattern of closed orbits as Θ→0.