21 research outputs found
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.Comment: Lecture given at International Conference on Quantum Mechanics and
Chaos, Osaka, September 200
Superscars
Wave functions of plane polygonal billiards are investigated. It is
demonstrated that they have clear structures (superscars) related with families
of classical periodic orbits which do not disappear at large energy
Random matrices associated with general barrier billiards
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-dimensional barrier billiards. These random matrices are
extracted from the exact billiard quantisation condition by applying a random
phase approximation for high-excited states. An important ingredient of the
method is the calculation of -matrix for the scattering in the slab with a
half-plane inside by the Wiener-Hopf method. It appears that these random
matrices have the form similar to the one obtained by the author in
[arXiv:2107.03364] for a particular case of symmetric barrier billiards but
with different choices of parameters. The local correlation functions of the
resulting random matrices are well approximated by the semi-Poisson
distribution which is a characteristic feature of various models with
intermediate statistics. Consequently, local spectral statistics of the
considered barrier billiards is (i) universal for almost all values of
parameters and (ii) well described by the semi-Poisson statistics.Comment: 28 pages, 6 figure
Quantum and Arithmetical Chaos
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 of the Laplace--Beltrami operator for the modular
group. The lectures cover a wide range of quantum chaos applications and can
serve as a non-formal introduction to mathematical methods of quantum chaos.Comment: Lectures given at Les Houches School "Frontiers in Number Theory,
Physics and Geometry" March 200
Statistical properties of structured random matrices
Spectral properties of Hermitian Toeplitz, Hankel, and Toeplitz-plus-Hankel
random matrices with independent identically distributed entries are
investigated. Combining numerical and analytic arguments it is demonstrated
that spectral statistics of all these random matrices is of intermediate type,
characterized by (i) level repulsion at small distances, (ii) an exponential
decrease of the nearest-neighbor distributions at large distances, (iii) a
non-trivial value of the spectral compressibility, and (iv) the existence of
non-trivial fractal dimensions of eigenvectors in Fourier space. Our findings
show that intermediate-type statistics is more ubiquitous and universal than
was considered so far and open a new direction in random matrix theory.Comment: 34 pages, 7 figure
BIOREGULATORY SYSTEMS AND HUMAN HEALTH
The subject of the study is the transition processes of the regulatory systems of the human body during bioresonance therapy. The purpose of the work is to study the transient processes in various regulatory systems of the human body during bioresonance effects on the body using the Brimton hardware and software complex. Analysis of the transient processes during the impact on the human body during BRT allows us to establish the moment of “harmonization” of the studied body systems and objectify the assessment of its condition.Key words: bioresonance therapy, transition processes, heterogeneity index, system, regulation
Formation of superscar waves in plane polygonal billiards
International audiencePolygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices. It is demonstrated that in the semiclassical limit multiple singular scattering on such vertices when optical boundaries of many scatters overlap leads to vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to singular scattering on boundary vertices, waves propagated inside periodic orbit pencils in the semiclassical limit tend to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes (or doorway states). For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space