Statistics of multiple avoided crossings

Abstract

The author considers the energy levels of a generic quantum mechanical system with Hamiltonian H depending on a parameter X. If the energy levels E_n are plotted as a function of X, the curves do not cross. The points of closest approach are called avoided crossings; these have a distinctive geometry and are important because they determine the limits of applicability of the adiabatic theorem. He describes some theoretical results on the parameter space density of avoided crossings, for systems with the spectral statistics of the Gaussian orthogonal ensemble (GOE). These results are in good agreement with numerical experiments.