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Level compressibility of certain random unitary matrices
Eugene Bogomolny
The value of spectral form factor at the origin, called level compressibility, is an important characteristic of random spectra. The paper is devoted to analytical calculations of…
Random matrices associated with general barrier billiards
Eugene Bogomolny
The paper is devoted to the derivation of random unitary matrices whose spectral statistics is the same as statistics of quantum eigenvalues of certain deterministic two-dimensiona…
Barrier billiard and random matrices
Eugene Bogomolny
The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonst…
Calculation of multifractal dimensions in spin chains
Y. Y. Atas, E. Bogomolny
It was demonstrated in [Phys. Rev. E 86, 021104, (2012)], that the ground-state wave functions for a large variety of one-dimensional spin-1/2 models are multifractals in the natur…
Integrable random matrix ensembles
E. Bogomolny, O. Giraud, C. Schmit
We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The const…
Random matrix ensembles associated with Lax matrices
E. Bogomolny, O. Giraud, C. Schmit
A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain dist…
Near integrable systems
E. Bogomolny, M. R. Dennis, R. Dubertrand
A two-dimensional circular quantum billiard with unusual boundary conditions introduced by Berry and Dennis (\emph{J Phys A} {\bf 41} (2008) 135203) is considered in detail. It is…
Trace formula for dieletric cavities : I. General properties
E. Bogomolny, R. Dubertrand, C. Schmit
The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as…
Distance matrices and isometric embeddings
E. Bogomolny, O. Bohigas, C. Schmit
We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenval…
Random wave functions and percolation
E. Bogomolny, C. Schmit
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…
First Experimental Observation of Superscars in a Pseudointegrable Barrier Billiard
E. Bogomolny, B. Dietz, T. Friedrich +4
With a perturbation body technique intensity distributions of the electric field strength in a flat microwave billiard with a barrier inside up to mode numbers as large as about 70…
Spectral statistics of a quantum interval-exchange map
E. Bogomolny, C. Schmit
Curious spectral properties of an ensemble of random unitary matrices appearing in the quantization of a map p -> p+alpha, q -> q+f(p+alpha) in [Giraud et al. nlin.CD/0403033] are…
Superscars
Eugene Bogomolny, Charles Schmit
Wave functions of plane polygonal billiards are investigated. It is demonstrated that they have clear structures (superscars) related with families of classical periodic orbits whi…
Quantum and Arithmetical Chaos
Eugene Bogomolny
The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the two-point spectral correlation functions of Riemann zeta function zeros, and…
Multiplicities of Periodic Orbit Lengths for Non-Arithmetic Models
Eugene Bogomolny, Charles Schmit
Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain grou…
Asymptotic behaviour of multiple scattering on infinite number of parallel demi-planes
Eugene Bogomolny, Charles Schmit
The exact solution for the scattering of electromagnetic waves on an infinite number of parallel demi-planes has been obtained by J.F. Carlson and A.E. Heins in 1947 using the Wien…
Spectral properties of distance matrices
E. Bogomolny, O. Bohigas, C. Schmit
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…
Nearest-neighbor distribution for singular billiards
E. Bogomolny, O. Giraud, C. Schmit
The exact computation of the nearest-neighbor spacing distribution P(s) is performed for a rectangular billiard with point-like scatterer inside for periodic and Dirichlet boundary…
Percolation model for nodal domains of chaotic wave functions
E. Bogomolny, C. Schmit
Nodal domains are regions where a function has definite sign. In recent paper [nlin.CD/0109029] it is conjectured that the distribution of nodal domains for quantum eigenfunctions…
Semi-classical calculations of the two-point correlation form factor for diffractive systems
E. Bogomolny, O. Giraud
The computation of the two-point correlation form factor K(t) is performed for a rectangular billiard with a small size impurity inside for both periodic or Dirichlet boundary cond…
Singular statistics
E. Bogomolny, U. Gerland, C. Schmit
We consider the statistical distribution of zeros of random meromorphic functions whose poles are independent random variables. It is demonstrated that correlation functions of the…
Spectral statistics and periodic orbits
E. Bogomolny
The main purpose of these lectures is to discuss briefly recent methods of calculation of statistical properties of quantum eigenvalues for chaotic systems based on semi-classical…