Quantum Chaos of the Hadamard-Gutzwiller Model

Abstract

We present a theory of quantum chaos of the Hadamard-Gutzwiller model, a quantum mechanical system which describes the motion of a particle on a surface of constant negative curvature. The theory is based on periodic-orbit sum rules that can be rigorously derived from the Selberg trace formula and which provide an exact substitute, appropriate for our strongly chaotic system, for the Bohr-Sommerfeld-Einstein quantization rules. Our recent enumeration of the classical periodic orbits enables us to evaluate the sum rules numerically and to demonstrate thereby that the theory provides also a practical method to study the quantum chaos of spectra.

Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics

“…iv) The conjecture has been checked numerically for several integrable (like the isospectral billiard shown in Figure 1.5) and chaotic systems [88,93] and has been found to hold with high statistical significance. v) In Figure 1.4 we show the numerical evaluation of f (α) for the strongly chaotic Hadamard-Gutzwiller model [64] which is the quantum version of the For the numerical computation in Figure 1.4 we used the first 6000 eigenvalues with positive parity (computed by the boundary-element method [97]) of a generic (nonarithmetic) Riemann surface whose fundamental domain in the Poincaré-disk model for hyperbolic geometry is described in [97]. We conclude that the computed histogram is in nice agreement with the conjecture (1.73).…”

“…Today the quantum system governed by the free Schrödinger equation i.e. the eigenvalue problem of the Laplace-Beltrami operator on these hyperbolic manifolds (or orbifolds), is known as the Hadamard-Gutzwiller model [64,65,107]. In dimension d = 3, hyperbolic manifolds are possible candidates for the spatial section of the Universe and are investigated in cosmology [108].…”

“…The torus T 2 is a compact Riemann surface of genus 1 and Gaussian curvature K = 0. A generalization to surfaces of higher genus is given by the famous Selberg trace formula [45,46] which has been much studied in the field of quantum chaos (see for example [47,[64][65][66]) and string theory (see for example [67,68]) and will be discussed in Section 1.4.4. The trace formula (1.37), respectively (1.42), has the typical structure of a trace formula and is in some sense a "meta formula" since it allows one to derive an infinite number of relations depending on the special choice of the spectral function h( p) satisfying the conditions (1.36).…”

Quantum and Arithmetical Chaos

“…Nevertheless when the first large scale numerical calculations were performed [3], [52] they clearly indicated that for certain hyperbolic models the spectral statistics were quite close to Poisson statistics typical for integrable systems.…”

Space–time approach to quantum chaos

“…I add that a large number of systems is known for which the first three terms of (2) describe the mean behaviour of NðEÞ very well even down to the ground state (see, for example, Fig. 2 in [12], Fig. 11 in [13], Fig.…”