Periodic-orbit theory of level correlations

Area of Science:

• Quantum Chaos
• Statistical Mechanics
• Mathematical Physics

Background:

• The Bohigas-Giannoni-Schmit conjecture proposes universality in spectral fluctuations for chaotic dynamical systems.
• Understanding these fluctuations is key to bridging quantum mechanics and classical chaos.

Purpose of the Study:

• To provide a semiclassical explanation for the Bohigas-Giannoni-Schmit conjecture.
• To derive the universal spectral correlator using semiclassical methods.

Main Methods:

• Utilized a generating function approach in the semiclassical limit.
• Analyzed quadruplets of sets of periodic orbits to determine the function's behavior.
• Employed asymptotic expansions for both nonoscillatory and oscillatory components of the spectral correlator.

Main Results:

• Obtained asymptotic expansions for the universal spectral correlator.
• Demonstrated that Borel summation of the derived series precisely reproduces the random-matrix theory correlator.

Conclusions:

• The semiclassical method successfully explains the universality of spectral fluctuations in chaotic dynamics.
• This work connects periodic orbit theory with random-matrix theory predictions for spectral statistics.