Related papers: Quantization of Sinai's Billiard - A Scattering Ap…
The semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived. The parametric dependence of the spectrum on the…
We present a comprehensive semiclassical investigation of the three-dimensional Sinai billiard, addressing a few outstanding problems in "quantum chaos". We were mainly concerned with the accuracy of the semiclassical trace formula in two…
The semiclassical description of billiard spectra is extended to include the diffractive contributions from orbits which are nearly tangent to a concave part of the boundary. The leading correction for an unstable isolated orbit is of the…
For the first time a three--dimensional (3D) chaotic billiard -- the 3D Sinai billiard -- was quantized, and high--precision spectra with thousands of eigenvalues were calculated. We present here a semiclassical and statistical analysis of…
We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared to the wave length 2 pi/k. Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant.…
We study the spectral rigidity problem for Sinai billiards with finite horizon, specifically asking whether the geometry of the billiard table can be recovered from the lengths of its (marked) periodic trajectories. To address this, we…
The methods of the high energy semiclassical quantization in the rational polygon billiards used in our earlier papers are generalized to an arbitrary rational multi-connected polygon billiards i.e. to the billiards which is a rational…
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using…
The semiclassical trace formula provides the basic construction from which one derives the semiclassical approximation for the spectrum of quantum systems which are chaotic in the classical limit. When the dimensionality of the system…
We present an efficient method to solve scattering problems in two-dimensional open billiards with two leads and a complicated scattering region. The basic idea is to transform the scattering region to a rectangle, which will lead to…
In billiard systems with a flux line semiclassical approximations for the density of states contain contributions from periodic orbits as well as from diffractive orbits that are scattered on the flux line. We derive a semiclassical…
A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measure preserving as a…
We estimate the error in the semiclassical trace formula for the Sinai billiard under the assumption that the largest source of error is due to Penumbra diffraction, that is diffraction effects for trajectories passing within a distance R…
The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which…
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…
The structure of the semiclassical trace formula can be used to construct a quasi-classical evolution operator whose spectrum has a one-to-one correspondence with the semiclassical quantum spectrum. We illustrate this for marginally…
We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of…
In the tight-binding approximation we consider multi-channel transmission through a billiard coupled to leads. Following Dittes we derive the coupling matrix, the scattering matrix and the effective Hamiltonian, but take into account the…
We present a semiclassical approximation to the scattering wavefunction $\Psi(\mathbf{r},k)$ for an open quantum billiard which is based on the reconstruction of the Feynman path integral. We demonstrate its remarkable numerical accuracy…
For classical billiards we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As…