Computationally induced "irregularity" in the spectra of integrable quantum systems.

Abstract

The nearest-neighbor spacing distributions P(S) for the variationally determined eigenvalues of two separable model Hamiltonians (two-dimensional quartic oscillators) are studied as functions of the dimension n of the variational basis. For large basis sizes the spacing histograms for a given energy range can be well fitted by a Poisson distribution, reflecting the regularity of the spectrum of a completely separable Hamiltonian. The decrease of n leads to histograms of qualitatively different spacing (for the same energy range as before), which can be well fitted by Brody distributions ${P}_{q}$(S). The Brody parameter q controls the transition between a Poisson (q=0) and a Wigner distribution (q=1). For large n, q=0 holds. When n falls below a critical value, q=1 results, i.e., ${P}_{q}$(S) turns over into a Wigner distribution. The latter is assumed to characterize a completely irregular spectrum in the classically chaotic energy range of a nonseparable Hamiltonian. The transition between a regular and an irregular spectrum is shown to be induced only by a change of numerical accuracy. We conjecture that similar behavior should also be observed in nonseparable systems.