Quantum-classical correspondence in polygonal billiards

Abstract

We show that wave functions in planar rational polygonal billiards (all angles rationally related to π) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly related to the classical invariant surfaces in the semiclassical limit. This is illustrated for the barrier billiard. We expect that these states are also present in integrable billiards with point scatterers or magnetic flux lines.

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…The dynamics on such a surface of higher genus is not quasiperiodic, which is reflected by multifractal Fourier spectra of dynamical variables (Artuso et al, 2000;Wiersig, 2000). Moreover, the quantum-classical correspondence is exotic (Bogomolny and Schmit, 2004;Wiersig, 2001) and the quantum spectrum obeys critical statistics (Bogomolny et al, 1999;Wiersig, 2002).…”

Hexagonal dielectric resonators and microcrystal lasers

“…However, there are some peculiar features that distinguish these billiards strongly from integrable ones (and also from fully or partially chaotic billiards): (i) an invariant surface does not have the topology of a torus but instead that of a surface of higher genus [4], loosely speaking, a torus with additional handles; (ii) the dynamics is not quasiperiodic. This is reflected, for example, by multifractal Fourier spectra of classical observables [7,8]; (iii) the quantum-classical correspondence is exotic [9]; (iv) the quantum spectrum obeys critical statistics [10]. To compute the quantum spectrum of a pseudointegrable billiard with a semiclassical treatment is extremely difficult, if possible at all.…”

Robust asymptotic tracking of periodic trajectories in elliptical billiards

“…wherex e is the nominal part of x e , i.e.,x e = x dẋdx T a T withx a (t) being defined in (7) and it has jumps at times t d i , i ∈ Z, i > 0 (see (11)). From (8), (5) and (6) the dynamics ofx e during the free-motion can be easily obtained aṡ…”

“…Such a system is called an elliptical billiards. The notion of billiard system was introduced by Birkhoff in [10], and since then it became a challenging research field, which has attracted the attention of researchers from mathematics, engineering and physics, where billiards are used to investigate the transition from quantum mechanics to classical mechanics (see, e.g., [11]). A lot of work has been done to study the properties of trajectories followed by a free particle on billiards of different shape and dimension, that is when no control is exerted on the moving mass (see, e.g., [12]).…”