Convergence of the scar series for the unsmoothed spectral Wigner function of a classically chaotic system (Berry, 1989)

In his 1989 phase-space theory of quantum scars, M. V. Berry expressed the epsilon-smoothed spectral Wigner function $W(\mathbf{x};E,\epsilon)$ of a classically chaotic system as an ergodic energy-surface term plus a sum of 'scar' contributions, each a fringe pattern decorating an isolated closed classical orbit, with amplitude $1/\sqrt{\det(M+I)}$ (M the linearized Poincaré map) and exponential damping $e^{-\epsilon T/\hbar}$ (T the orbit period). Whether this scar series survives the removal of the smoothing — the limit $\epsilon\to0$, in which $W$ resolves individual eigenstates — is not established: the closed-orbit sum is known to diverge for the spectral density of systems with exponentially proliferating long orbits (e.g. geodesic motion on surfaces of constant negative curvature), and the divergence is more acute for the phase-space function $W$ itself, whose scar terms carry orbit-length-growing factors. The problem is to determine whether the superposition of scars can represent the highly oscillatory Wigner functions of individual chaotic eigenstates, whether a convergent or resummed (finite) form of the phase-space scar series exists, or whether a minimum smoothing $\epsilon$ is required for convergence; this is the phase-space eigenfunction analogue of the eigenvalue-side resummation problem solved only formally for the spectral determinant.

Background

For a quantum system whose classical Hamiltonian motion is chaotic, the Gutzwiller trace formula — the semiclassical expression of the density of states as a sum over classical periodic orbits — is an asymptotic, not convergent, series: the number of orbits grows exponentially with period, and each term carries stability factors that defeat naive summation. The analogous question for eigenstates is sharper. Heller (1984) discovered numerically that individual eigenfunctions of classically chaotic systems show enhanced density along unstable periodic orbits ('scars'), contradicting the naive semiclassical expectation (the semiclassical eigenfunction hypothesis) that chaotic eigenstates are locally random superpositions of plane waves uniformly covering the energy surface $H(\mathbf{x})=E$, where $\mathbf{x}=(\mathbf{q},\mathbf{p})$ is a phase-space point and $H$ the Hamiltonian. Bogomolny (1988) derived smoothed semiclassical wavefunctions showing oscillations on closed orbits, but with spatial and energy smoothing. The natural phase-space object is the spectral Wigner function $W(\mathbf{x};E,\epsilon)=h^N\sum_n \delta_\epsilon(E-E_n)W_n(\mathbf{x})$: the sum over eigenstates $n$ of their Wigner functions $W_n$ (the Wigner function is the standard phase-space quasi-distribution of a quantum state) weighted by a Lorentzian of width $\epsilon$ centered at energy $E$; $h$ is Planck's constant, $\hbar=h/2\pi$, and $N$ the number of degrees of freedom. Berry (1989) computed the scar contribution of an isolated closed orbit of period $T$ and action $S=\oint \mathbf{p}\cdot d\mathbf{q}$: near the orbit, with transverse Poincaré-section coordinates $\mathbf{X}=(\mathbf{Q},\mathbf{P})$ and linearized Poincaré map $\mathbf{X}_B=M\mathbf{X}_A$ (the linearized return map of the transverse section onto itself after one traversal of the orbit), the scar is a cosine in $\mathbf{X}$ with transverse fringe spacing of order $\hbar^{1/2}$, multiplied by an Airy function of $H(\mathbf{x})-E$ describing fringes of spacing $\hbar^{2/3}$ perpendicular to the energy surface, with overall factor $2^N/\sqrt{\det(M+I)}$ and damping $e^{-\epsilon T/\hbar}$; the Maslov-type phase $\gamma$ accounts for caustics (folds at which the family of trajectories tangent to the orbit fails to project single-valuedly onto position space). Integrated over phase space, the scar term reproduces the corresponding Gutzwiller density-of-states term, and for long orbits the amplitude decays as $e^{-h_{\mathrm{KS}}T/2}$ with $h_{\mathrm{KS}}$ the Kolmogorov–Sinai entropy (the sum of positive Lyapunov exponents, which measures the rate of separation of neighboring trajectories). The smoothing $\epsilon$ is essential to the derivation: it regularizes the sum over orbits by suppressing long ones, exactly as energy averaging regularizes the trace formula. In the classical limit the scar terms become weak ($O(\hbar^{N-1})$) delta-function concentrations on the closed orbits, on top of the dominant term concentrated on the energy surface. Berry's paper left the convergence structure of this construction open: for the unsmoothed spectral density, the closed-orbit sum is known to diverge for systems with exponentially proliferating orbits (e.g. geodesic motion on surfaces of constant negative curvature) because the proliferation of orbits outruns the decay of the stability factor $\det(M-I)^{-1/2}$ in the classical-limit scar expression, and for the phase-space function $W$ the divergence is worse since the scar terms themselves contain orbit-length-growing factors; an attempt to enforce consistency of the series through an operator identity had not succeeded. A companion gap, stated in the same paper, concerns the refined semiclassical theory of the dominant energy-surface term, specifically its Airy fringes near classically forbidden regions, which the author deferred to later publication. On the eigenvalue side, the divergence of periodic-orbit sums is a recognized fundamental problem, addressed by Keating (1992), whose formal Riemann–Siegel-type resummation of the spectral determinant converts the orbit sum into a finite one for computing energy levels, and by Eckhardt and Russberg (1993), who studied zeta-function and Padé resummation of cycle expansions for spectra; no analogous resummation is known for the phase-space scar series.

Problem Statement

Determine the convergence structure of Berry's scar series for the spectral Wigner function of a classically chaotic system in the limit of vanishing smoothing, $\epsilon\to0$. Specifically: (i) establish whether the representation of $W(\mathbf{x};E,\epsilon)$ as the ergodic energy-surface term plus the sum over closed-orbit scar contributions admits a well-defined limit as $\epsilon\to0$ — possibly after a resummation — or whether the series necessarily diverges and requires a minimum smoothing $\epsilon$ (with its dependence on $\hbar$ and $E$) to converge; and (ii) decide, as a consequence, whether a superposition of scar contributions can represent the full oscillatory phase-space structure of the Wigner functions of individual eigenstates, or only of energy-averaged states. The question concerns generic chaotic systems with exponentially proliferating closed orbits, for which the orbit-stability factors $\det(M-I)^{-1/2}$ (and, in the scar terms, factors growing with orbit length) make the raw series formally divergent; the joint roles of the damping factor $e^{-\epsilon T/\hbar}$, orbit period $T$, and $\hbar$ must be controlled. Resolving parts (i) and (ii) together resolves the task. A connected secondary component from the same source — the completion of the refined semiclassical theory of the dominant energy-surface term, including its Airy fringes near classically forbidden regions, beyond the rigorized smoothed-sum asymptotics now available — is optional.

References

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