C. Schmit

Spectral properties of distance matrices

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 spectrum exhibits delocalized and strongly localized states that possess different power-law average behaviour. The exponents depend only on the dimensionality of the manifold.

Distribution of Eigenvalues for the Modular Group

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 they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.

Semiclassical computations of energy levels

Different methods of semiclassical calculations of energy levels of two-dimensional ergodic models are discussed and compared. Special attention is given to the calculation of the dynamical zeta function via the Rieman-Siegel relations.

Anomalous spectral statistics in a symmetrical billiard

It is a common assumption that time reversal invariance or some other anti-unitary symmetry for a classically chaotic system implies GOE-type spectral fluctuations for the corresponding quantum system. Based on previous work on structural invariance, we show that a time-reversal invariant system for certain point symmetries displays the GUE statistics typical of systems with broken time-reversal symmetry; specifically a billiard having only three-fold symmetry shows this unexpected behaviour.

Multiplicities of Periodic Orbit Lengths for Non-Arithmetic Models

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 groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length. The main ingredient used is the construction of a joint distribution of periodic orbits when group matrices are transformed by field isomorphisms. The method can be generalized to other groups for which traces of group matrices are integers of an algebraic field of finite degree.

Asymptotic behaviour of multiple scattering on an infinite number of parallel half-planes

The exact solution for the scattering of electromagnetic waves on an infinite number of parallel half-planes was obtained by J F Carlson and A E Heins in 1947 using the Wiener–Hopf method (Carlson J F and Heins A E 1947 Quart. Appl. Math. 4 313–79). We analyse their solution in the semiclassical limit of small wavelength and find the asymptotic behaviour of the reflection and transmission coefficients. The results are compared with those obtained within the Kirchhoff approximation.

Spectral statistics of a pseudo-integrable map: the general case*

It is well established numerically that spectral statistics of pseudo-integrable models differs considerably from the reference statistics of integrable and chaotic systems. In Bogomolny and Schmit (2004 Phys. Rev. Lett. 93 254102) statistical properties of a certain quantized pseudo-integrable map had been calculated analytically but only for a special sequence of matrix dimensions. The purpose of this paper is to obtain the spectral statistics of the same quantum map for all matrix dimensions.