Spectral Staircase Research Articles

Exact and approximate energy sums in potential wells

Sums of the N lowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regime N ≫ 1. Euler-Maclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levels N + 1, N + 2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10−7. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on how N and the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits.

Spectral and parametric averaging for integrable systems

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos, spectral averaging (SA) and parametric averaging (PA). For SA, we introduce a new procedure, namely, rescaled spectral averaging (RSA). Unlike traditional SA, it can describe the correlation function of spectral staircase (CFSS) and produce persistent oscillations of the interval level number variance (IV). PA while not as accurate as RSA for the CFSS and IV, can also produce persistent oscillations of the global level number variance (GV) and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.

FLUCTUATIONS OF THE RIEMANN STAIRCASE

The one-sided power spectrum of the fluctuations of the Riemann staircase is calculated numerically and shown to be Lorentzian, like the one-sided power spectrum of the fluctuations of the spectral staircase for classically chaotic systems. This result provides further evidence in support of Berry's conjecture that the En's of the enigmatic Riemann zeroes 1/2 + iEn are real because they are the quantized energies of a system which is classically chaotic.

Isospectral shapes with Neumann and alternating boundary conditions.

The isospectrality of a well-known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the Neumann boundary condition is verified numerically to high precision. Equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating Dirichlet and Neumann boundary conditions along successive edges. Good agreement with theory is obtained for the corresponding spectral staircase functions. Strong numerical evidence is also presented for isospectrality in an example of a different pair of shapes whose basic building-block triangle is not isosceles. Some possible confirmatory experiments involving fluids are suggested.

Semiclassical Trace Formulae and Eigenvalue Statistics in Quantum Chaos

A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local spectral statistics, measuring correlations among finitely many eigenvalues, are reviewed and a detailed semiclassical analysis of the number variance is given. Thereafter the transition to global spectral statistics, taking correlations among infinitely many quantum energies into account, is discussed. It is emphasized that the resulting limit distributions depend on the way one passes to the global scale. A conjecture on the distribution of the fluctuations of the spectral staircase is explained in this general context and evidence supporting the conjecture is discussed.

Thermodynamical fingerprints of fractal spectra

We investigate the thermodynamics of model systems exhibiting two-scale fractal spectra. In particular, we present both analytical and numerical studies on the temperature dependence of the vibrational and electronic specific heats. For phonons, and for bosons in general, we show that the average specific heat can be associated to the average (power law) density of states. The corrections to this average behavior are log-periodic oscillations which can be traced back to the self-similarity of the spectral staircase. In the electronic case, even if the thermodynamical quantities exhibit a strong dependence on the particle number, regularities arise when special cases are considered. Applications to substitutional and hierarchical structures are discussed.

Universal Signatures in the Mode Fluctuation Distribution of Chaotic Systems

An example other than the hyperbolic billiard, the mode fluctuation of which shows Gaussian distribution, is presented. It is the Bunimovich's stadium billiard where the mode fluctuation is obtained by eliminating the semiclassical contribution of the bouncing ball modes from the spectral staircase function. The mode fluctuation without this elimination shows the skewness which considerably agrees with semiclassically evaluated predictions.

Aspects of Quantum Chaos in Generic Systems

Hamiltonian dynamical systems with a small number of degrees of freedom, and having ergodic classical dynamics, exhibit universal statistical laws as for their spectral fluctuations of quantal energy levels around the (non-universal) smooth average of the spectral staircase. This finding is one of the cornerstones of the stationary problems in quantum chaos: The so-called Bohigas-Giannoni-Schmit conjecture [16] states that in such cases the level statistics of the random matrix theories (of GOE if there is an antiunitary symmetry in the system, or of GUE otherwise) apply. This conjecture has been supported by many numerical experiments, but also by the semiclassical results of Berry [7], and it has recently been announced to be proven by Andreev et al [1], but under stronger conditions than ergodicity (exponential decay of classical correlations). In cases of classical integrability we also find universality, namely the statistics is typically Poissonian, although there are exceptions, whose measure somehow is expected to be zero or, at least, very small. Finally, there is the class of systems with mixed classical dynamics (regular motions coexisting in the classical phase space with chaotic regions), which are dynamically generic systems, in which case a semiclassical theory can be set up [10, 67, 53, 54]. We present recent results on this class of Hamiltonian systems.

Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

This paper is devoted to the quantum chaology of three-dimensional systems. A trace formula is derived for compact polyhedral billiards which tessellate the three-dimensional hyperbolic space of constant negative curvature. The exact trace formula is compared with Gutzwiller's semiclassical periodic-orbit theory in three dimensions, and applied to a tetrahedral billiard being strongly chaotic. Geometric properties as well as the conjugacy classes of the defining group are discussed. The length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase N( E), which in turn is successfully applied in a quantization condition.

QUANTIZATION RULES FOR STRONGLY CHAOTIC SYSTEMS

We discuss the quantization of strongly chaotic systems and apply several quantization rules to a model system given by the unconstrained motion of a particle on a compact surface of constant negative Gaussian curvature. We study the periodic-orbit theory for distinct symmetry classes corresponding to a parity operation which is always present when such a surface has genus two. Recently, several quantization rules based on periodic orbit theory have been introduced. We compare quantizations using the dynamical zeta function Z(s) with the quantization condition [Formula: see text] where a periodic-orbit expression for the spectral staircase N(E) is used. A general discussion of the efficiency of periodic-orbit quantization then allows us to compare the different methods. The system dependence of the efficiency, which is determined by the topological entropy τ and the mean level density [Formula: see text], is emphasized.

Level statistics of a noncompact integrable billiard.

A noncompact, nonintegrable billiard of importance in cosmology---the infinite equilateral triangle on a space of constant negative curvature---gives rise to a natural noncompact integrable approximation, which is studied here quantum mechanically. The Weyl formula for the averaged spectral staircase contains a nonstandard logarithmic term due to the noncompactness of the billiard. For two symmetry classes we determine numerically the spectral staircase, the level-spacing distribution p(s), and the averaged ${\mathrm{\ensuremath{\Delta}}}_{3}$ statistic, and compare with analytical expressions that follow from Poisson statistics and semiclassical theory. The Poissonian result for the correlation function of ${\mathrm{\ensuremath{\Delta}}}_{3}$ is derived in closed form and also compared with the data.

Refined asymptotic expansion for the partition function of unbounded quantum billiards

This paper presents a refined asymptotic expansion for the partition function Θ(t)=Tr etΔ of quantum billiards in the unbounded regions {0≤x,0≤yxμ≤1}, μ>0, and {0≤ye‖x‖≤1}⊆R2, where Δ is the Dirichlet Laplacian. Simon [Ann. Phys. 146, 209 (1983); J. Funct. Anal. 53, 84 (1983)] determined the leading divergence of the trace of the heat kernel for the first class of systems. Standard techniques are combined for the evaluation of Θ for bounded region billiards with results by Van den Berg [J. Funct. Anal. 71, 279 (1987)] for ‘‘horn-shaped regions’’ using an optimized way of dividing the region into ‘‘narrow’’ and ‘‘wide’’ parts to determine the first three terms in the asymptotic expansion of Θ. Results are also stated for bounded regions with cusps that can be obtained by the same method. As an application, the spectral staircase of the strongly chaotic billiard system defined in the region {0≤xy≤2,x≥0}, which has been discussed in connection with the Riemann ζ function and the search for quantum chaos is considered.

Neutrino billiards: time-reversal symmetry-breaking without magnetic fields

A Dirac hamiltonian describing massless spin-half particles (‘neutrinos’) moving in the plane r = ( x, y ) under the action of a 4-scalar (not electric) potential V(r) is, in position representation, H ^ = − i h c σ ^ ⋅ ∇ + V ( r ) σ ^ z , , where σ̂ = (σ̂ x , σ̂ y ) and σ̂ z are the Pauli matrices; Ĥ acts on two-component column spinor wavefunctions ψ ( r ) = ( ψ 1 , ψ 2 ) and has eigen­values ћck n . Ĥ does not possess time-reversal symmetry ( T ). If V ( r ) describes a hard wall bounding a finite domain D (‘billiards’), this is equivalent to a novel boundary condition for ψ 2 / ψ 1 . T -breaking is interpreted semiclassically as a difference of π between the phases accumulated by waves travelling in opposite senses round closed geo­desics in D with odd numbers of reflections. The semiclassical (large- k ) asymptotics of the eigenvalue counting function (spectral staircase) N ( k ) are shown to have the ‘Weyl’ leading term Ak 2 /4π, where A is the area of D, but zero perimeter correction. The Dirac equation is transformed to an integral equation round the boundary of D, and forms the basis of a numerical method for computing the k n . When D is the unit disc, geodesics are integrable and the eigenvalues, which satisfy J l ( k n ) = J l +1 ( k n ), are (locally) Poisson-distributed. When D is an ‘Africa’ shape (cubic conformal map of the unit disc), the eigenvalues are (locally) distributed according to the statistics of the gaussian unitary ensemble of random-matrix theory, as predicted on the basis of T -breaking and lack of geometric symmetry.

Semiclassical theory of spectral rigidity

The spectral rigidity ⊿( L ) of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of L mean level spacings. In the semiclassical limit (ℏ→0), formulae are obtained giving ⊿( L ) as a sum over classical periodic orbits. When L ≪ L max , where L max ~ℏ-(N-1) for a system of N freedoms, ⊿( L ) is shown to display the following universal behaviour as a result of properties of very long classical orbits: if the system is classically integrable (all periodic orbits filling tori), ⊿( L )═ 1 / 5 L (as in an uncorrelated (Poisson) eigenvalue sequence); if the system is classically chaotic (all periodic orbits isolated and unstable) and has no symmetry, ⊿( L ) ═ In L /2π 2 + D if 1≪ L ≪ L max (as in the gaussian unitary ensemble of random-matrix theory); if the system is chaotic and has time-reversal symmetry, ⊿( L ) = In L /π 2 + E if 1 ≪ L ≪ L max (as in the gaussian orthogonal ensemble). When L ≫ L max , ⊿( L ) saturates non-universally at a value, determined by short classical orbits, of order ℏ –(N–1) for integrable systems and In (ℏ -1 ) for chaotic systems. These results are obtained by using the periodic-orbit expansion for the spectral density, together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere. For two examples ⊿(L) is studied in detail: the rectangular billiard (integrable), and the Riemann zeta function (assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic).