Quantum chaos in one dimension?

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→∞ the solution is nowhere differentiable and most probably nowhere continuous. Thus such a counterexample does not exist.