Some recursive formulas for Selberg-type integrals

A set of recursive relations satisfied by Selberg-type integrals involving monomial symmetric polynomials are derived, generalizing previously known results. These formulas provide a well-defined algorithm for computing Selberg-Schur integrals whenever the Kostka numbers relating Schur functions and the corresponding monomial polynomials are explicitly known. We illustrate the usefulness of our results discussing some interesting examples.Comment: 11 pages. To appear in Jour. Phys.

The Selberg trace formula for Dirac operators

We examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to non-trivial multiplicities of eigenvalues. The relation to spectra of Maass-Laplace operators is also exploited. Our main result is a Selberg trace formula for Dirac operators on hyperbolic surfaces

Random matrix theory, the exceptional Lie groups, and L-functions

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groupsComment: 14 page

Relativistic Wavepackets in Classically Chaotic Quantum Cosmological Billiards

Close to a spacelike singularity, pure gravity and supergravity in four to eleven spacetime dimensions admit a cosmological billiard description based on hyperbolic Kac-Moody groups. We investigate the quantum cosmological billiards of relativistic wavepackets towards the singularity, employing flat and hyperbolic space descriptions for the quantum billiards. We find that the strongly chaotic classical billiard motion of four-dimensional pure gravity corresponds to a spreading wavepacket subject to successive redshifts and tending to zero as the singularity is approached. We discuss the possible implications of these results in the context of singularity resolution and compare them with those of known semiclassical approaches. As an aside, we obtain exact solutions for the one-dimensional relativistic quantum billiards with moving walls.Comment: 18 pages, 10 figure

Some extremal functions in Fourier analysis, III

We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\lambda|x|}$, where $\lambda >0$, $\log |x|$ and $|x|^{\alpha}$, where $-1 < \alpha < 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.Comment: 26 pages. Submitte

Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface \mM with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0.

A new proof of the Vorono\"i summation formula

We present a short alternative proof of the Vorono\"i summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schr\"odinger operator on a non-compact quantum graph \mathfrak{G} [S. Egger n\'e Endres and F. Steiner, J. Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new proof of a non-trivial identity for a particular Lambert series which involves the divisor function d(n) and is identical with the trace of the Euclidean wave group of the Laplacian on the infinite graph \mathfrak{G}.Comment: Enlarged version of the published article J. Phys. A: Math. Theor. 44 (2011) 225302 (11pp

Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.Comment: 19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of cusped manifolds in section 2 is correcte

Finite N Fluctuation Formulas for Random Matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - )$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (\theta_j - \pi)$ and $- \sum_{j=1}^N \log 2|\sin \theta_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty

Bohr-Sommerfeld Quantization of Periodic Orbits

We show, that the canonical invariant part of $\hbar$ corrections to the Gutzwiller trace formula and the Gutzwiller-Voros spectral determinant can be computed by the Bohr-Sommerfeld quantization rules, which usually apply for integrable systems. We argue that the information content of the classical action and stability can be used more effectively than in the usual treatment. We demonstrate the improvement of precision on the example of the three disk scattering system.Comment: revte