Spectral statistics of the circle billiard with a point scatterer: exact spacing law for affected levels, simultaneous Bessel zeros, rationality effects, and genericity of smooth location dependence

A circular billiard of radius R containing a single point scatterer at $r_0 = \alpha R$ is a canonical integrable system with a localized perturbation. Rahav, Richman and Fishman (J. Phys. A 36, L529, 2003) showed that in the semiclassical limit its spectrum splits into two uncorrelated sequences: an 'unaffected' Poisson sequence (scatterer in the classically forbidden region, of relative density $X_{un} = (2/\pi)[\arccos(\alpha) - \alpha\sqrt{1-\alpha^2}]$) and an 'affected' sequence, and they derived a Berry–Robnik-type nearest-neighbour spacing distribution (NNSD) for the superposed spectrum by assuming — without analytical justification, and known to fail for at least one global spectral measure — a semi-Poisson law for the affected levels. The letter's four unresolved issues and their subsequent literature: the exact NNSD of the affected sequence remains unknown; the question whether one scatterer location can make infinitely many symmetric (in $\phi$) eigenfunctions vanish simultaneously — equivalently whether the fixed-ratio coincidence system $J_m(kR)=0=J_{m+l}(k_1 R)$, $J_m(\alpha k R)=0=J_{m+l}(\alpha k_1 R)$, pairing eigenfunctions of distinct orders that both vanish at the scatterer, admits infinitely many solutions for one and the same $\alpha \in (0,1)$ — is open, as is the density of near-coincidences relevant to possible rationality-induced peculiarities; and the conjecture that smooth dependence of the statistics on scatterer location is typical of integrable systems with localized perturbations is unproven, with only qualitative experimental support. All four issues remain open.

Background

Quantum chaos predicts universal energy-level statistics according to the nature of the classical dynamics. For classically integrable systems (dynamics constrained on invariant tori) the unfolded levels (energies monotonically rescaled so that the local mean spacing is one) obey Poisson statistics (Berry and Tabor 1977), with nearest-neighbour spacing distribution (NNSD) $P(S) = e^{-S}$ showing no level repulsion; for strongly chaotic time-reversal-symmetric systems they obey Wigner–Dyson random-matrix statistics of the Gaussian orthogonal ensemble (GOE — the ensemble of real symmetric matrices with independent Gaussian entries) (Bohigas, Giannoni and Schmit 1984), with $P(S)$ vanishing linearly at small spacing $S$ (level repulsion). Here $S$ is the gap between consecutive levels in units of the mean level spacing. When two statistically independent level sequences of relative densities $\rho$ and $1-\rho$ are superposed, the resulting NNSD is of Berry–Robnik type (Berry and Robnik 1984).

Between these limits lie systems with intermediate statistics. A canonical family is an integrable billiard perturbed by a point scatterer — a zero-range (formally delta-function) impurity at an interior point, defined rigorously by self-adjoint extension theory (a one-parameter family of boundary conditions at the impurity that replaces the ill-defined literal two-dimensional delta potential). The perturbed eigenvalues are the zeros of a secular equation built from the Green function of the unperturbed system, schematically $1 + \lambda \sum_n |\psi_n(\mathbf{x}_0)|^2/(E-E_n)$, where $\lambda$ is a renormalized coupling and $|\psi_n(\mathbf{x}_0)|^2$ the unperturbed eigenfunction intensity at the scatterer position $\mathbf{x}_0$; only states with non-negligible amplitude there are shifted. The archetype is the Šeba billiard (Šeba 1990), a rectangle with a point scatterer. Its statistics are intermediate — shifted levels repel each other — and depend sensitively on the rationality of the scatterer's coordinates relative to the side lengths, because infinitely many rectangular eigenfunctions vanish simultaneously at rational points. Bogomolny, Gerland and Schmit (2001) developed the 'singular statistics' of such systems — the statistics of zeros of random meromorphic functions with Poisson-distributed poles — computing two-point correlations analytically; Bogomolny, Giraud and Schmit (2002) computed the exact NNSD for the rectangular case and found it of intermediate type, but not the semi-Poisson and not the Poisson law. A widely used phenomenological stand-in is the semi-Poisson distribution $P(S) = 4S e^{-2S}$, with linear level repulsion and an exponential tail; it fits numerics reasonably in several pseudointegrable and point-scatterer systems (e.g. Wiersig and Carlo 2003 for a truncated triangle) but has no analytic justification, and it fails for at least one global measure: the spectral form factor (the Fourier transform of the two-level correlation function) at zero delay equals 1/2 for semi-Poisson but 1 for point-scatterer billiards (Rahav and Fishman 2002).

The circular billiard is integrable: Dirichlet eigenfunctions are $J_m(kr)\cos(m\phi)$ and $J_m(kr)\sin(m\phi)$ with $J_m$ the Bessel function of the first kind, eigenvalues fixed by $J_m(kR) = 0$, and a two-fold $\pm m$ degeneracy. Classically, trajectories are chords conserving angular momentum $L$, with a caustic (turning point) at minimal radius $r_{min} = L/\sqrt{2mE}$, where $m$ here is the particle mass (not the angular-momentum quantum number $m$ used to label the quantum states). Placing a point scatterer at $(r_0, \phi = 0)$, $r_0 = \alpha R$, and restricting to the sector symmetric in $\phi$ (antisymmetric states vanish at the scatterer and decouple), Rahav, Richman and Fishman (2003) proved a semiclassical split: states whose classical turning point exceeds $r_0$ (scatterer in the classically forbidden region, where the wavefunction is exponentially small by the WKB (Wentzel–Kramers–Brillouin) semiclassical approximation) are essentially unshifted, while the others are affected. The unaffected fraction is $X_{un} = (2/\pi)[\arccos(\alpha) - \alpha\sqrt{1-\alpha^2}]$, varying smoothly from 1 (scatterer at the centre) to 0 (at the boundary). Assuming Poisson statistics for the unaffected sequence and a semi-Poisson NNSD for the affected sequence, Berry–Robnik superposition gives a closed full NNSD that agrees well with numerical eigenvalues at indices 40000–60000 for $\alpha = 0.2, 0.6, 0.9$, with deviations for $\alpha$ close to 1 attributed to the affected-sequence NNSD differing from semi-Poisson. The letter leaves open: the exact affected-sequence statistics; whether many symmetric eigenfunctions can vanish simultaneously at one scatterer location — since a state vanishes at the scatterer iff $J_m(\alpha k R) = 0$ when $J_m(kR) = 0$, this is a fixed-ratio coincidence question among zeros of Bessel functions; whether near-coincidences of such zeros (which must exist, since $J_m$ is asymptotically a cosine) produce weak rationality-type peculiarities analogous to the rectangle; and whether the smooth dependence of the statistics on scatterer location is typical of integrable systems with localized perturbations.

Problem Statement

The problem is to complete the quantitative theory of the intermediate spectral statistics of a circular billiard of radius $R$ containing a single point scatterer at radius $r_0 = \alpha R$ ($0 < \alpha < 1$) on the ray $\phi = 0$, in the semiclassical (high-eigenvalue-index) limit, in the sector symmetric in $\phi$ — and to settle the genericity of the observed behaviour for integrable systems with localized perturbations. Rahav, Richman and Fishman (2003) established the semiclassical split into an unaffected sequence (classical turning point $r_{min} > r_0$; Poisson; fraction $X_{un} = (2/\pi)[\arccos(\alpha) - \alpha\sqrt{1-\alpha^2}]$) and an affected sequence, and derived the superposed nearest-neighbour spacing distribution (NNSD) only under a semi-Poisson ansatz for the affected sequence. The unresolved issues, in the letter's own framing, are:

(1) Exact affected-level statistics. Determine the exact NNSD of the affected level sequence, replacing the semi-Poisson ansatz $P_2(S) = 4(1-X_{un})^2 S e^{-2(1-X_{un})S}$, which has no analytical justification, is known to be inconsistent with the expected form-factor value at zero delay (1 for point-scatterer systems versus 1/2 for semi-Poisson), and leaves unexplained the observed deviations of the full NNSD from the predicted Berry–Robnik superposition for $\alpha$ close to 1.

(2) Simultaneous zeros of Bessel functions. Determine whether there exists a scatterer location $\alpha \in (0,1)$ at which infinitely many distinct symmetric (in $\phi$) unperturbed eigenfunctions vanish simultaneously. Writing $j_{m,n}$ for the $n$-th positive zero of the Bessel function $J_m$, a symmetric state of order $m$ and radial index $n$ vanishes at the scatterer precisely when $J_m(\alpha j_{m,n}) = 0$, i.e. when $\alpha = j_{m,n'}/j_{m,n}$ for some zero $j_{m,n'}$ of the same function $J_m$; the letter poses the question as whether the system $J_m(kR) = 0 = J_{m+l}(k_1 R)$ and $J_m(\alpha k R) = 0 = J_{m+l}(\alpha k_1 R)$ — two eigenfunctions of distinct orders $m$ and $m+l$ (integers $m \ge 1$, $l \ge 1$) that both vanish at the scatterer — admits infinitely many solutions for one and the same $\alpha$. Then determine the density, as a function of $\alpha$, of the near-zeros that must exist because $J_m$ is asymptotically a cosine, and decide whether that density is large enough to affect the spectral statistics: determine whether the resulting shift of the full NNSD or of the spectral form factor is resolvable at the eigenvalue ranges already computed (indices $10^4$–$10^5$), and whether it persists or decays as the eigenvalue index grows.

(3) Rationality-induced peculiarities. Determine whether scatterer locations of special arithmetic form — reduced rationals $\alpha = p/q$ in $(0,1)$ with $q \le 20$ — yield spectral statistics measurably differing — in the manner of the rational-scatterer-position anomalies of the rectangular (Šeba) billiard — from the smooth $X_{un}$/Berry–Robnik description and from nearby non-arithmetic locations, at eigenvalue indices up to $N = 10^6$; 'measurably differing' means a deviation of the full NNSD or of the spectral form factor that exceeds the statistical uncertainty of the sample over that index range. This location-resolved comparison is (3)'s own target: it is not settled by (2)'s generic near-coincidence density or by (2)'s persistence-versus-decay question, since neither resolves the arithmetic dependence of the statistics on the scatterer location.

(4) Genericity of smooth location dependence. Establish or refute the conjecture that the smooth dependence of the spectral statistics on the scatterer location — governed by the continuous fraction $X_{un}$ — is typical of integrable systems with localized perturbations, while the rationality-sensitive behaviour of the rectangle is atypical. Here the ambient class is integrable planar billiards carrying a single interior point scatterer (of which the circle and the rectangle are members); 'typical' means that the smooth $X_{un}$-governed dependence holds without fine-tuning the scatterer location to special arithmetic values, and 'atypical' means that the rationality-sensitive behaviour occurs only at such special values. Determine whether, within this class, the rectangle's arithmetic sensitivity is exceptional and no member depends on the scatterer location non-smoothly for a reason other than an arithmetic special value, or whether a further member of the class has point-scatterer statistics that vary non-smoothly with the scatterer location for a reason other than an arithmetic special value.

These four issues are the unresolved questions left by the source letter and share one objective — a complete, quantitative account of the intermediate statistics of the point-perturbed circular billiard and of its genericity. Issues (2) and (3) are the structural questions underlying the accuracy of the semi-Poisson ansatz in (1) (exact or near simultaneous vanishing of many eigenfunctions at the scatterer is precisely the mechanism by which rectangle-like anomalies would enter), and (4) is the generalization the letter conjectures.

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