Quantum mushroom billiards

We report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. A. Bunimovich [Chaos 11, 802 (2001)]. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E(-1/3) (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16,000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers (approximately 10(5)) orders of magnitude faster than with competing methods.     

Quantum aspects of triangular billiards

The energy spectrum of a particle enclosed in a two-dimensional triangular box can in general not be calculated analytically. Therefore, in order to establish the degree of intergrability, using some well known criteria, this spectrum must be calculated numerically.

For more than 800 triangles these calculations were performed and then used to check a number of conjectures found in the literature. For almost all of them we find counter examples.

Our own suggestion is that in general acute triangles are more integrable than obtuse ones. However, also this rule has its exceptions, which cannot be explained away by some triangles being more rational than others.     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Quantum billiards in the shape of right triangles

We have studied quantum properties of right triangular billiards as a function of their acute angle . The energy spectrum and wavefunction plots with the nodal pattern are provided; symmetry arguments are given for regularities found in the classically integrable cases =30° and 45°. A comparison of the level spacing distributions for different confirms the tendency towards level repulsion in the nonintegrable domain.     

Quantum transport in lattices of coupled electronic billiards

A new system with dynamic chaos—2D lattice of single Sinai billiards coupled through quantum dots—is studied experimentally. Localization in such a system was found to be substantially suppressed, because the characteristic size of the billiard for g≤1 (g is conductance measured in e ₂/h units) is the localization length rather than the de Broglie wavelength of an electron, as in the usual 2D electron system. Lattice ballistic effects (commensurate peaks in the magnetoresistance) for g?1, as well as extremely large magnetoresistance caused by the interference in chaotic electron trajectories, were found. Thus, this system is shown to be characterized by simultaneous existence of effects that are inherent in order (commensurate peaks of magnetoresistance), disorder (percolation charge transport), and chaos (weak localization in chaotic electron trajectories).     

Free-field representation of permutation branes in Gepner models

We consider a free-field realization of Gepner models based on the free-field realization of N = 2 superconformal minimal models. Using this realization, we analyze the A/B-type boundary conditions starting from the ansatz with the left-moving and right-moving free-field degrees of freedom glued at the boundary by an arbitrary constant matrix. We show that the only boundary conditions consistent with the singular vector structure of unitary minimal model representations are given by permutation matrices, thereby yielding an explicit free-field construction of the permutation branes of Recknagel. The text was submitted by the author in English.     

Quantum noncommutative multidimensional cosmology

We present exact quantum solutions for a noncommutative, multidimensional cosmological model and show that stabilization of extra dimensions sets in with the introduction of noncommutativity between the scale factors. An interpretation is offered to accommodate the notion of time, rendering comparison with the classical solutions possible.     

Quantum chaos in nano-sized billiards in layered two-dimensional semiconductor structures

We consider two-dimensional, electron-rich cavities that can be created at a (AlGa)As-GaAs interface. In the modelling of such cavities we include features that are typical for small semiconductor structures or devices, i.e., soft walls representing electrostatic confinement and disorder due to ionized impurities. The introduction of soft walls is found to have a profound effect on the dynamic behaviour. There are situations in which there is a crossover from a Wigner distribution for the nearest level spacing to an effectively Poisson-like one as the confining walls are softened. The crossover occurs in a region which is accessible experimentally. A mechanism for the crossover is discussed in terms of groups of energy levels being separated from each other as walls become soft. The effects of disorder are found to be negligible for high-mobility samples, i.e., the motion of the particles is ballistic. These findings are of a general nature. Chaotic Robnik dots, circular dots with a special "dent," are also investigated. In this case there is no crossover from Wigner to Poisson distributions. An explanation for this difference is proposed. Finally, the effects of leads are investigated in an elementary way by simply attaching two stubs to a circular dot. For wide stubs, which in our simple model would correspond to open leads, we obtain Wigner statistics indicating a transition to irregular behaviour. A lead-induced transition of this kind appears consistent with recent measurements of the line-shape of the weak localization peak, observed in the low-temperature magnetoresistance of square semiconductor billiards. Finally, implications for conductance fluctuations are briefly commented on. (c) 1996 American Institute of Physics.     

Quantum creation of multidimensional universes

A non zero probability amplitude for the appearance of a multidimensional universe of (1+d) dimensions is found. This can happen either in a “symmetric phase”, in which all dimensions are in an exponential expansion, or else in a “broken phase”, withd ₁ dimensions inflating exponentially andd ₂ forming a sphere of constant radius. The value of these amplitude for different total number of dimensions is discussed, and so physical consequences for Kaluza-Klein cosmologies are drawn.     

Quantum chaos of bogoliubov waves for a bose-einstein condensate in stadium billiards

We investigate the possibility of quantum (or wave) chaos for the Bogoliubov excitations of a Bose-Einstein condensate in billiards. Because of the mean field interaction in the condensate, the Bogoliubov excitations are very different from the single particle excitations in a noninteracting system. Nevertheless, we predict that the statistical distribution of level spacings is unchanged by mapping the non-Hermitian Bogoliubov operator to a real symmetric matrix. We numerically test our prediction by using a phase shift method for calculating the excitation energies.     

Dynamical reduction in multidimensional cosmological models

We present conditions under which there occurs a dynamical dimensional reduction of cosmological models in the form of Bianchi I×(N-3)-dimensional torus filled with matter of the ideal fluid type.     

Quantum effects in cosmological models with singularities

The vacuum expectation values of the energy-momentum tensor of quantized scalar and spinor fields in a de Sitter space of the first kind are calculated. Limiting cases of the obtained exact expressions are considered. It is noted that the de Sitter space is a self-consistent solution of the Einstein equations with allowance for quantum vacuum fluctuations of massless fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 67–70, January, 1981.I thank V. M. Mostepanenko and B. N. Sharapov for numerous helpful discussions.