One-Dimensional Conservative System with Quantum Level Statistics of Gaussian Orthogonal Ensembles

Abstract

We numerically study quantum mechanics of one-dimensional (1D) time-independent system whose energy level statistics obeys the Gaussian orthogonal ensemble. ID conservative systems are known to be integrable. However, at least numerically, it is also shown that we can construct the potential for the Schrodinger equation that reproduces a finite number of given energy levels of chaotic regime, e.g., the random matrix theory. In this work a potential is constructed numerically by the standard gradient method. The more energy levels of chaotic regime we take, the more complicated and finer the ripples of the potential become. Then the potential has fractal structure at high energy limit and its fractal dimension is determined to be d = 1.7.