Semiclassical Origin of Quantum Shells in the Elliptic Billiard

Holger Waalkens, Jan Wiersig, Holger R. Dullin

Holger Waalkens

Rudi Schäfer, Ulrich Kuhl, Michael Barth +1 more

Barbara Dietz

Quantum signatures of classical chaos in the spectral properties of non-relativistic quantum billiards are well understood by now based on the semiclassical and the random matrix theory aproach.The experimental detection of relativistic phenomena featured by graphene implicated an increasing interest in the properties of graphene billiards and relativistic quantum billiards consisting of massless spin-1/2 particles governed by the Dirac equation and confined to a planar domain.Confinement is attained by imposing appropriate boundary conditions on the wave function components.We chose those proposed for neutrino billiards.By now numerical studies have been performed on the spectral properties of neutrino billiards of various shapes.Our main focus is the question to what extent the semiclassical approach in terms of trace formulae applies to such systems.We derive trace formulae for circular and elliptical neutrino billiards.The associated eigenvalue equations have been solved analytically for the circular billiard.We present analytical solutions for elliptical neutrino billiards.

Xu Xue-You, Yanhui Zhang, Huang Fa-zhong +3 more

The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.

Holger Waalkens, Jan Wiersig, Holger R. Dullin

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 The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and