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Quantum and Arithmetical Chaos

Eugene Bogomolny

TL;DR

The work surveys quantum chaos through trace formulas, emphasizing the exact Selberg trace formula for hyperbolic surfaces and its Riemann-zeta counterpart. It develops semiclassical tools (Green functions, Gutzwiller formula) to connect quantum spectra with classical periodic orbits, and extends these ideas to integrable, chaotic, and arithmetic systems. A central theme is the universality of spectral statistics across models, contrasted with the peculiar Poisson-like behavior in arithmetic cases due to exponential orbit-length degeneracies, and the deep links between primes, zeta functions, and quantum spectra. The study highlights both rigorous trace-formula results and heuristic approaches (Hardy–Littlewood, RMT conjectures), underscoring the rich interplay between physics, geometry, and number theory. The findings illuminate how arithmetic structure can dramatically alter quantum fluctuations even in classically chaotic settings, with implications for understanding universality and spectral correlations in complex systems.

Abstract

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.

Quantum and Arithmetical Chaos

TL;DR

The work surveys quantum chaos through trace formulas, emphasizing the exact Selberg trace formula for hyperbolic surfaces and its Riemann-zeta counterpart. It develops semiclassical tools (Green functions, Gutzwiller formula) to connect quantum spectra with classical periodic orbits, and extends these ideas to integrable, chaotic, and arithmetic systems. A central theme is the universality of spectral statistics across models, contrasted with the peculiar Poisson-like behavior in arithmetic cases due to exponential orbit-length degeneracies, and the deep links between primes, zeta functions, and quantum spectra. The study highlights both rigorous trace-formula results and heuristic approaches (Hardy–Littlewood, RMT conjectures), underscoring the rich interplay between physics, geometry, and number theory. The findings illuminate how arithmetic structure can dramatically alter quantum fluctuations even in classically chaotic settings, with implications for understanding universality and spectral correlations in complex systems.

Abstract

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.

Paper Structure

This paper contains 63 sections, 517 equations, 20 figures, 1 table.

Table of Contents

  1. Trace Formulas
  2. Plane Rectangular Billiard
  3. More Refined Approach
  4. Example
  5. Billiards on Constant Negative Curvature Surfaces
  6. Hyperbolic Geometry
  7. Discrete groups
  8. Example
  9. Modular Group
  10. Fundamental Region
  11. Classical Mechanics
  12. Quantum Problem
  13. Construction of the Green Function
  14. Density of State
  15. Mean Density of States
  16. ...and 48 more sections

Figures (20)

  • Figure 1: The trace formula for the rectangular billiard with periodic boundary conditions calculated by taking into account 250 different periodic orbits. Dotted lines indicate the position of exact energy levels.
  • Figure 2: The Poincaré model of constant negative curvature space. Solid line indicates the geodesic passing through points A and B.
  • Figure 3: Fundamental domain of the modular group. The indicated parts are identified by the corresponding generators
  • Figure 4: Fundamental domain of multiplication group
  • Figure 5: Zeros of the Selberg zeta function
  • ...and 15 more figures