Quantum Mechanical Approach to Bifurcation Point Detection in Hamiltonian Dynamical Systems

TL;DR

This paper introduces a quantum mechanical method using two-point correlation functions to detect bifurcation points in Hamiltonian dynamical systems, demonstrated through numerical analysis of a lemon-shaped billiard.

Contribution

It presents a novel quantum-based approach for identifying bifurcation points by analyzing spike oscillations in the two-point correlation function.

Findings

Spike oscillations in TPCL indicate bifurcation points.

Abrupt increases in reduced chi-squared value detect bifurcations.

Method successfully applied to lemon-shaped billiard system.

Abstract

Energy level statistics of a bounded quantum system, whose classical dynamical system exhibits bifurcations, is investigated using the two-point correlation function (TPCL), which at the bifurcation points exhibits periodic spike oscillations owing to the accumulation of levels called the shell effect. The spike oscillations of the TPCL is analyzed by the reduced chi-squared value which deduced to exhibit abrupt increases at bifurcation points, thereby yielding a novel detection approach. Using this method, we attempt to numerically detect the bifurcation points of a lemon-shaped billiard.