Elliptic Quantum Billiard

Abstract

The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the amplitude and form of the different kinds of peaks is analytically calculated.

Quantum-classical correspondence in polygonal billiards

“…3 resembles strongly that of integrable systems with separatrices [23][24][25]. It is therefore not surprising that the statistical properties of the mean energies of |L -states are similar to those of energy levels of integrable systems.…”

Quantized conductance through an asymmetric narrow constriction in a three-dimensional electron gas

“…In the elliptical cross section in the x-y plane the caustic of BB 2 , which consists of two hyperbolas resulting from = s 1 , is that of the "bouncing ball modes" which one finds in the billiard in a planar ellipse. 26 In contrast to that the motion in phase BB 1 , though oscillatory in and , does not touch the boundary hyperboloid, i.e., the corresponding toroidal cylinders are foliated by straight lines of free motions without reflections. A pair ͑s 1 2 , s 2 2 ͒ in phase BB 3 represents motion which does not cross the x-y plane.…”

“…The separated Helmholtz equations corresponding to the transverse degrees of freedom become the separated wave equations in a planar elliptic quantum billiard. 26 This is most easily seen from the wave equations in ͑10͒. We therefore define new variables = + K͑qЈ͒ and = + K͑q͒ and new separation constants k = k / a 2 and l= l / a 2 .…”

A difference-equation formalism for the nodal domains of separable billiards

“…2.1c). The quantum numbers r and l are defined so as to be consistent with the Einstein-Brillouin-Keller (EBK) quantization for the symmetry reduced system [29]. For the elliptical annulus, defined as the regions D = {(ρ, φ) : ρ min ≤ ρ ≤ ρ max }, we follow a procedure analogous to the case of the circular annuli.…”