Transition in eigenvalue statistics due to tunneling in a simple quantum system

Statistical properties of the sequence of energy eigenvalues in a quantum system are related to the classical dynamics of the system. Sequences of energy levels for classically regular systems generally exhibit Poisson statistics, while those for classically chaotic systems exhibit random matrix statistics. For many systems the classical dynamics can change from regular to chaotic as the result of changing a system parameter and the quantum system shows a corresponding change in its eigenvalue statistics. Here we present a simple quantum system, composed of an infinite square well with several Dirac delta barriers placed inside it, that exhibits a change in eigenvalue statistics as the transmission probability T through the delta barriers changes. For T≈0 the distribution of level spacings is Poisson-like, but increasing T causes the distribution to change to match that of the Gaussian Orthogonal Ensemble. For even greater T values the level spacing distribution becomes Gaussian with a standard deviation that approaches zero as T→1. A similar transition in statistics is seen in the number variance for the eigenvalue sequence. We also demonstrate that the structure of the energy eigenstates changes from highly localized at T≈0 to evenly spread across the infinite square well as T→1. Our model system can be viewed as an example of a quantum graph, and we show that our results match with the expectations for quantum graphs in the limit T→0.