For a bound quantum system whose classical phase space at fixed energy contains both regular islands (motion on invariant tori) and chaotic components, Berry and Robnik (1984) derived an explicit interpolation formula for the nearest-neighbor level-spacing distribution P(S) by superposing statistically independent Poisson sequences from the regular regions and Wigner (Gaussian-orthogonal-ensemble, GOE) sequences from the chaotic regions, weighted by the Liouville measures of the corresponding phase-space regions. The derivation rests on three unproven ingredients — Wigner/GOE statistics for chaotic components, Poisson statistics and independence for regular regions, and the neglect of tunnelling between classically separated regions — and its 1984 numerical testing was infeasible. Later work has confirmed the formula numerically on scaling billiards in the semiclassical limit and has quantified tunnelling corrections at small spacings, but the central ingredients remain without first-principles derivation: a proof that a connected classically chaotic component yields GOE spacing statistics (the core of the Bohigas–Giannoni–Schmit conjecture), a rigorous derivation of the dynamical-tunnelling corrections to statistical independence, and an analysis of whether cantori (partial barriers) and sticky orbits spoil the Poisson character of the regular sequence. The problem is to place the Berry–Robnik statistics on a rigorous semiclassical foundation, deriving each ingredient and the formula's domain of validity.

Background

In quantum chaos one studies how the statistical properties of a bound quantum system's energy spectrum reflect the nature of the underlying classical dynamics. After subtracting the smooth average level density (a procedure called unfolding), the nearest-neighbor spacing distribution P(S)P(S) is a standard diagnostic: for classically integrable systems, whose trajectories lie on invariant tori in phase space, Berry and Tabor (1977) showed that P(S)P(S) is Poisson, P(S)=exp⁡(−S)P(S)=\exp(-S) (mean spacing normalized to one), reflecting the absence of level repulsion; for classically chaotic (ergodic) systems, overwhelming numerical evidence (Bohigas, Giannoni and Schmit, 1984) shows that P(S)P(S) follows the Wigner surmise of random-matrix theory, P(S)=(π/2)Sexp⁡(−πS2/4)P(S)=(\pi/2)S\exp(-\pi S^2/4), characteristic of the Gaussian orthogonal ensemble (GOE) of real symmetric random matrices. There is no rigorous proof of this universality for generic chaotic systems — this is the content of the Bohigas–Giannoni–Schmit (BGS) conjecture; semiclassical periodic-orbit theory (the Gutzwiller trace formula, which expresses the quantum density of states as a sum over classical periodic orbits) reproduces spectral correlations only in controlled limits, via the diagonal approximation, in which only the terms pairing each periodic orbit with itself are kept in the double orbit sum (Berry, 1985), and its off-diagonal extensions to correlated orbit pairs (Bogomolny and Keating, 1996).

Generic Hamiltonian systems, however, are neither integrable nor fully chaotic: at a given energy EE the classical phase space is mixed, consisting of islands of regular motion on tori embedded in a chaotic sea. Berry and Robnik (1984) proposed that in the semiclassical limit ℏ→0\hbar\to 0 the spectrum is the statistically independent superposition of Poisson sequences from the regular regions (with total level density ρ1\rho_1, proportional to the Liouville measure of the regular part of the energy surface) and Wigner sequences from each connected chaotic region (level densities ρi\rho_i proportional to their Liouville measures; the Liouville measure is the natural phase-space volume normalized so that the whole energy surface has measure one). This yields, with total density ρ=∑iρi\rho=\sum_i\rho_i, the formula

P(S)=1ρd2dS2[e−ρ1S∏i=2Nerfc⁡(π2ρiS)],P(S)=\frac{1}{\rho}\frac{d^2}{dS^2}\left[e^{-\rho_1 S}\prod_{i=2}^{N}\operatorname{erfc}\left(\frac{\sqrt{\pi}}{2}\rho_i S\right)\right],

where erfc is the complementary error function. For a single chaotic component this gives a one-parameter interpolation family between Poisson and Wigner statistics indexed by the regular fraction ρ1\rho_1.

Berry and Robnik themselves stated the assumptions on which the formula rests. First, in 1984 the predicted P(S)P(S) could not be numerically tested on the standard mixed models (such as coupled anharmonic oscillators), because holding the classical phase-space structure fixed while accumulating enough levels requires an impracticably small ℏ\hbar; they suggested scaling systems such as planar billiards, where the classical phase space is energy-independent, as the proper testing ground. Second, the assumed statistical independence of the level sequences from different regions neglects quantum tunnelling across classically closed boundaries between islands and the chaotic sea (dynamical tunnelling); they argued the effect is confined to exponentially small spacings S∼exp⁡(−const/ℏ)S\sim\exp(-\mathrm{const}/\hbar), invisible on the semiclassical spacing scale, but gave no rigorous justification. Third, the assignment of Wigner/GOE statistics to each connected chaotic component rests on numerical evidence and on Pechukas's (1983) argument about the level dynamics of irregular spectra, not on a derivation from first principles. Fourth, collecting the infinitely many individual regular tori into a single Poisson sequence generalizes the integrable-system result, but the possible effects of incomplete tori (cantori — remnants of broken tori that act as partial transport barriers in phase space) and of sticky orbits (chaotic trajectories trapped for long times near island boundaries) were not addressed.

Problem Statement

Establish the Berry–Robnik level-spacing statistics for bound quantum systems with mixed classical phase space on a rigorous first-principles footing, by deriving each ingredient of the superposition formula and the formula's domain of validity. Specifically: (1) establish (prove or derive) that a connected classically chaotic component of a mixed energy surface produces a Wigner (Gaussian-orthogonal-ensemble, GOE) nearest-neighbor spacing distribution for its share of the quantum spectrum in the semiclassical limit — the ingredient inherited from the unproven Bohigas–Giannoni–Schmit conjecture; (2) derive from first principles the corrections to the assumed statistical independence of the level sequences from different phase-space regions caused by dynamical tunnelling across classically closed boundaries, quantifying the small-spacing behavior of P(S)P(S) and the sense in which tunnelling effects are negligible on the semiclassical spacing scale; and (3) determine the effect of cantori (partial barriers from broken tori) and sticky orbits on the Poisson character and independence of the regular level sequence entering the superposition. The 1984 numerical-testing obstacle is already resolved for scaling systems such as planar billiards, and the derivations above are to be consistent with the numerically confirmed Berry–Robnik distribution in the semiclassical limit and with its observed finite-ℏ\hbar deviations.

References

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