Quantum mechanical approach to bifurcation point detection in Hamiltonian dynamical systems

Abstract

Abstract The energy level statistics of a bounded quantum system whose classical dynamical system exhibits bifurcations is investigated using the two-point correlation function (TPCF), which at the bifurcation points exhibits periodic spike oscillations owing to the accumulation of levels called the shell effect. The spike oscillations of the TPCF are analyzed by the reduced chi-squared value, which exhibits 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.