E. Bogomolny

Distribution of the ratio of consecutive level spacings in random matrix ensembles.

We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.

Star graphs and Seba billiards

We derive an exact expression for the two-point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity. This turns out to be identical to the corresponding result for certain Seba billiards in the semiclassical limit. Reasons for this are discussed. The formula we derive is also shown to be equivalent to a series expansion for the form factor - the Fourier transform of the two-point correlation function - previously calculated using periodic orbit theory.We derive an exact expression for the two-point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity. This turns out to be identical to the corresponding result for certain eba billiards in the semiclassical limit. The reasons for this are discussed. The formula we derive is also shown to be equivalent to a series expansion for the form factor - the Fourier transform of the two-point correlation function - previously calculated using periodic orbit theory.

Circular dielectric cavity and its deformations

The construction of perturbation series for slightly deformed dielectric circular cavity is discussed in detail. The obtained formulas are checked on the example of cut disks. A good agreement is found with direct numerical simulations and far-field experiments.

Inferring periodic orbits from spectra of simply shaped microlasers

Dielectric microcavities are widely used as laser resonators and characterizations of their spectra are of interest for various applications. We experimentally investigate microlasers of simple shapes (Fabry-Perot, square, pentagon, and disk). Their lasing spectra consist mainly of almost equidistant peaks and the distance between peaks reveals the length of a quantized periodic orbit. To measure this length with a good precision, it is necessary to take into account different sources of refractive index dispersion. Our experimental and numerical results agree with the superscar model describing the formation of long-lived states in polygonal cavities. The limitations of the two-dimensional approximation are briefly discussed in connection with microdisks.

Eigenfunction entropy and spectral compressibility for critical random matrix ensembles.

Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D(1) and the spectral compressibility χ are related by the simple equation χ+D(1)/d=1, where d is system dimensionality.

SLE description of the nodal lines of random wavefunctions

The nodal lines of random wavefunctions are investigated. We demonstrate numerically that they are well approximated by the so-called SLE6 curves which describe the continuum limit of the percolation cluster boundaries. This result gives additional support to the recent conjecture that the nodal domains of random (and chaotic) wavefunctions in the semi-classical limit are adequately described by the critical percolation theory. It is also shown that using the dipolar variant of SLE reduces significantly finite size effects.

On the spacing distribution of the Riemann zeros: corrections to the asymptotic result

It has been conjectured that statistical properties of zeros of the Riemann zeta function near z = 1/2 + iE tend, as E → ∞, to the distribution of eigenvalues of large random matrices from the unitary ensemble. At finite E, numerical results show that the nearest-neighbour spacing distribution presents deviations with respect to the conjectured asymptotic form. We give here arguments indicating that to leading order these deviations are the same as those of unitary random matrices of finite dimension , where β = 1.573 14... is a well-defined arithmetic constant.

Directional emission of stadium-shaped microlasers

The far-field emission of two-dimensional stadium-shaped dielectric cavities is investigated. Microlasers with such shape present a highly directional emission. We provide experimental evidence of the dependence of the emission directionality on the shape of the stadium, in good agreement with ray numerical simulations. We develop an analytical geometrical optics model which permits to account for the main observed features. Wave numerical calculations confirm the results.

Trace formula for dielectric cavities. II. Regular, pseudointegrable, and chaotic examples.

Dielectric resonators are open systems particularly interesting due to their wide range of applications in optics and photonics. In a recent paper [Phys. Rev. E 78, 056202 (2008)] the trace formula for both the smooth and the oscillating parts of the resonance density was proposed and checked for the circular cavity. The present paper deals with numerous shapes which would be integrable (square, rectangle, and ellipse), pseudointegrable (pentagon), and chaotic (stadium), if the cavities were closed (billiard case). A good agreement is found between the theoretical predictions, the numerical simulations, and experiments based on organic microlasers.

Riemann Zeta Function and Quantum Chaos

A brief review of recent developments in the theory of the Riemann zeta function inspired by ideas and methods of quantum chaos is given.