Pseudointegrable Billiards Research Articles

Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard

We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.

Scale anomaly and quantum chaos in billiards with pointlike scatterers.

We argue that the random-matrix like energy spectra found in pseudointegrable billiards with pointlike scatterers are related to the quantum violation of scale invariance of classical analogue system. It is shown that the behavior of the running coupling constant explains the key characteristics of the level statistics of pseudointegrable billiards.

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width l in the common boundary. We show that such a system has at least one bound state for any l≳0. We find the corresponding eigenvalues and eigenfunctions numerically using the mode-matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase-shift spacing statistics shows that in distinction to closed pseudo-integrable billiards, the present system is essentially nonchaotic. Finally, we illustrate time evolution of wave packets in the present model.

Periodic orbits and spectral statistics of pseudointegrable billiards.

We demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than $l$ increases as $\frac{\ensuremath{\pi}{b}_{0}{l}^{2}}{〈a(l)〉}$, where ${b}_{0}$ is a constant and $〈a(l)〉$ is the average area occupied by these families. We also find that $〈a(l)〉$ increases with $l$ before saturating. Finally, we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies.

Non-universal spectral rigidity of quantum pseudo-integrable billiards

We obtain the Dyson - Mehta -statistic for pseudo-integrable billiards and show that it is non-universal with a universal trend, also that this trend is similar to the one for integrable billiards. We present a formula, based on exact semiclassical calculations and the proliferation law of periodic orbits, which gives rigidity for the entire range of L. To consolidate our theory, we discuss several examples finding complete agreement with the numerical results, and also the underlying fundamental reasons for the non-universality.

Avoided crossings: Curvature distribution and behavior of eigenfunctions of pseudointegrable and chaotic billiards.

We perform numerical studies on the curvature distribution of classically pseudointegrable and chaotic quantum billiards. This curvature distribution is Gaussian orthogonal ensemble--like for both systems and corresponds to linear level repulsion. The billiards show nonuniversal behavior at low curvatures. As an aside, avoided crossings of three levels are investigated for both systems. Strong interaction between the levels ${\mathit{E}}_{\mathit{i}}$ and ${\mathit{E}}_{\mathit{i}+2}$ is shown. Eigenvalues and eigenfunctions are calculated very precisely by a grid method with an asymmetric sparse matrix.

Statistical properties of spectra of pseudointegrable systems

We analyze the spectral properties of various quantum pseudointegrable billiards (rhombus, gnomon, deltoid) and link them to the genus of the invariant surface of the corresponding classical model. Numerical investigations of the quantum billiards are completed by an experimental study of microwave resonators. Absorption spectra of microwaves in ``ssL-shaped'' resonators are measured and the distributions of eigenfrequencies are investigated.

Fluctuations in the length spectrum of pseudo-integrable billiards

We derive expressions for the spectral rigidity Δ( L) of the length spectrum in pseudo-integrable billiards. Interestingly, we find that Δ( L) has a linear dependence, similar to that found in the integrable case. This is in marked contrast to the spectral rigidity of the energy spectrum, which is closer to that of chaotic systems.

Law of proliferation of periodic orbits in pseudointegrable billiards.

We prove that the periodic-orbit counting function, a measure of the rate of proliferation of periodic orbits, for a barrier billiard and the \ensuremath{\pi}/3-rhombus billiard is of the form ${\mathit{ax}}^{2}$+bx+c, where x is the length (equivalently, period) up to which periodic orbits are counted and a,b,c are system-specific constants. The generality of our arguments strongly suggests that the law of proliferation given here is a representation of general truth about two-dimensional plane-polygonal billiards.

Manifestation of wave chaos in pseudointegrable microwave resonators.

We analyze the distribution of eigenfrequencies of microwave resonators which correspond to pseudointegrable billiards. The statistical properties of the measured spectra are explained by means of a model of additive random matrices.

Signature of chaos in a quantum pseudointegrable billiard.

We analyze the quantum level statistics of the zero-obstacle-size limit of the pseudointegrable billiard While the nearest-neighbor-spacing distribution approaches to the Poissonian shape in the semiclassical limit, it is shown to become the Wigner distribution in low-energy region. This fact is interpreted as a signature of the chaotic motion generated by the quantum uncertainty.

Signature of Chaos in Quantum Pseudointegrable Billiard(New Trends in Nuclear Collective Dynamics)

Signature of Chaos in Quantum Pseudointegrable Billiard(New Trends in Nuclear Collective Dynamics)

Quantum level statistics of pseudointegrable billiards.

We study the spectral statistics of systems of two-dimensional pseudointegrable billiards. These systems are classically nonergodic, but nonseparable. It is found that such systems possess quantum spectra which are closely simulated by the Gaussian orthogonal ensemble. We discuss the implications of these results on the conjectured relation between classical chaos and quantum level statistics. We emphasize the importance of the semiclassical nature of any such relation.

Unphysical singularities in semiclassical level density expansions for polygon billiards

It is pointed out that the semiclassical periodic orbit expansion for the level density of a bounded quantum system may contain singular structure not found in the exact level density. This is illustrated for a particular system, a pseudointegrable polygon billiard. The author shows that here the periodic orbit expansion has singular structure on all scales of energy, and takes on negative as well as positive values. However, when sufficiently smoothed the expansion bears a good resemblance to the exact smoothed level density.