Quantum chaos in one dimension?

TL;DR

This paper explores the inverse of a famous conjecture in quantum chaos, demonstrating through numerical methods that the potential with eigenvalues following random matrix statistics is likely nowhere continuous, thus no such counterexample exists.

Contribution

It introduces two inversion methods to find one-dimensional potentials with eigenvalues obeying random matrix statistics, providing evidence against the existence of such potentials.

Findings

Potential is likely nowhere continuous in the limit

Counterexample potential does not exist

Numerical methods confirm theoretical predictions

Abstract

In this work we investigate the inverse of the celebrated Bohigas-Giannoni-Schmit conjecture. Using two inversion methods we compute a one-dimensional potential whose lowest N eigenvalues obey random matrix statistics. Our numerical results indicate that in the asymptotic limit, N->infinity, the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.