Eugène Bogomolny

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Nodal domains are regions where a function has definite sign. In [] it is conjectured that the distribution of nodal domains for quantum eigenfunctions of chaotic systems is universal. We propose a percolationlike model for description of these nodal domains which permits us to calculate all interesting quantities analytically, agrees well with numerical(More)
The two-point correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy– Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but(More)
Distance matrices are matrices whose elements are the relative distances between points located on a certain manifold. In all cases considered here all their eigenvalues except one are non-positive. When the points are uncorrelated and randomly distributed we investigate the average density of their eigenvalues and the structure of their eigenfunctions. The(More)
We consider the statistical distribution of zeros of random meromorphic functions whose poles are independent random variables. It is demonstrated that correlation functions of these zeros can be computed analytically, and explicit calculations are performed for the two-point correlation function. This problem naturally appears in, e.g., rank-1 perturbation(More)
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.(More)
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(More)
Matrices with random (or pseudo-random) elements appear naturally in different physical problems and their statistical properties have been thoroughly investigated (see e.g. [1]). A special case of random matrices, called distance matrices, has been recently proposed in [2]. They are defined for any metric space X with a probability measure μ on it as(More)
Recently it was conjectured that nodal domains of random wave functions are adequately described by critical percolation theory. In this paper we strengthen this conjecture in two respects. First, we show that, though wave function correlations decay slowly, a careful use of Harris’ criterion confirms that these correlations are unessential and nodal(More)
The 2-point correlation form factor, K2(τ), for small values of τ is computed analytically for typical examples of pseudo-integrable systems. This is done by explicit calculation of periodic orbit contributions in the diagonal approximation. The following cases are considered: (i) plane billiards in the form of right triangles with one angle π/n and (ii)(More)