Semiclassical Trace Formula Research Articles

A semi-classical trace formula at a non-degenerate critical level

We study the semi-classical trace formula at a critical energy level for a h-pseudodifferential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.

Space–time approach to quantum chaos

AbstractIn the first part of this paper, it is shown that the energy levels of a quantum system, whose classical limit is chaotic, encode certain space–time properties of the corresponding classical system. To see this, one considers the semiclassical limit as Planck's constant tends to zero. As a generalization of Mark Kac's famous question, it is demonstrated that “one can hear the periodic orbits of a quantum billiard”. In the second part, some mathematical aspects of the semiclassical limit are reviewed. In order to deal with expressions that are mathematically easier to control, one does not work with the path integral directly, but instead with a smoothed kernel corresponding to a well‐defined Fourier integral operator. Applying then the techniques from microlocal analysis and pseudodifferential operators, one arrives at a semiclassical trace formula which is a generalization of the Gutzwiller trace formula originally derived from the path integral. In the third part of this paper, the Hadamard–Gutzwiller model is discussed whose classical limit is a strongly chaotic (Anosov) system. In order to derive exact orbit sum rules for this model, one requires the path integral on hyperbolic D‐space (D ≥ 2) which can be exactly solved by using the general lattice definition of path integrals in curvilinear coordinates.

Analytical perturbative approach to periodic orbits in the homogeneous quartic oscillator potential

We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lam{\'e} functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical quantization. For the non-integrable situations, we show that the uniform contribution of the bifurcating period-one orbits to the coarse-grained density of states competes with that of the shortest isolated orbits, but decreases with increasing chaoticity parameter $\alpha$.

Spin phase-space semiclassics for weak spin orbit coupling

We apply the semiclassical spin coherent state method for the density of states by Pletyukhov et al (2002 Phys. Rev. Lett. 89 116601) in the weak spin–orbit coupling limit and recover the modulation factor in the semiclassical trace formula found by Bolte and Keppeler (1998 Phys. Rev. Lett. 81 1987; 1999 Ann. Phys., NY 274 125).

Semiclassical quantization with bifurcating orbits

Bifurcations of classical orbits introduce divergences into semiclassical spectra which have to be smoothed with the help of uniform approximations. We develop a technique to extract individual energy levels from semiclassical spectra involving uniform approximations. As a prototype example, the method is shown to yield excellent results for photoabsorption spectra for the hydrogen atom in an electric field in a spectral range where the abundance of bifurcations would render the standard closed-orbit formula without uniform approximations useless. Our method immediately applies to semiclassical trace formulas as well as the closed-orbit theory and offers a general technique for the semiclassical quantization of arbitrary systems.

Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. II: Spheroidal Cavity

We derive a semiclassical trace formula for the level density of the three-dimensional spheroidal cavity. To overcome the divergences and discontinuities occurring at bifurcation points and in the spherical limit, the trace integrals over the action-angle variables are performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell energies are in good agreement with quantum-mechanical results. We find that the births of three-dimensional orbits through the bifurcations of planar orbits in the equatorial plane lead to considerable enhancement of shell effect for superdeformed shapes.

Semiclassical trace formulae for systems with spin$ndash$orbit interactions: successes and limitations of present approaches

We discuss the semiclassical approaches for describing systems with spin–orbit interactions. We use these methods to derive trace formulae for several two- and three-dimensional model systems, and exhibit their successes and limitations. We also discuss, in particular, the mode conversion problem that arises in the strong-coupling limit.

Higher-order &plank; corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system.

Semiclassical calculations of the two-point correlation form factor for diffractive systems

The computation of the two-point correlation form factor K(τ) isperformed for a rectangular billiard with a small-size impurity inside forboth periodic or Dirichlet boundary conditions. It is demonstrated that allterms of perturbation expansion of this form factor in powers of τ canbe computed directly from a semiclassical trace formula. The main part of thecalculation is the summation of nondiagonal terms in the cross product ofclassical orbits. When the diffraction coefficient for the scattering on theimpurity is a constant our results coincide with the expansion of exactexpressions obtained earlier by a different method.

QUANTAL ANDREEV BILLIARDS: SEMICLASSICAL APPROACH TO MESOSCALE OSCILLATIONS IN THE DENSITY OF STATES

Andreev billiards are finite, arbitrarily-shaped, normal-state regions, surrounded by superconductor. At energies below the superconducting gap, single-quasiparticle excitations are confined to the normal region and its vicinity, the essential mechanism for this confinement being Andreev reflection. This Paper develops and implements a theoretical framework for the investigation of the short-wave quantal properties of these single-quasiparticle excitations. The focus is primarily on the relationship between the quasiparticle energy eigenvalue spectrum and the geometrical shape of the normal-state region, i.e. the question of spectral geometry in the novel setting of excitations confined by a superconducting pair-potential. Among the central results of this investigation are two semiclassical trace formulas for the density of states. The first, a lower-resolution formula, corresponds to the well-known quasiclassical approximation, conventionally invoked in settings involving superconductivity. The second, a higher-resolution formula, allows the density of states to be expressed in terms of: (i) An explicit formula for the level density, valid in the short-wave limit, for billiards of arbitrary shape and dimensionality. This level density depends on the billiard shape only through the set of stationary-length chords of the billiard and the curvature of the boundary at the endpoints of these chords; and (ii) Higher-resolution corrections to the level density, expressed as a sum over periodic orbits that creep around the billiard boundary. Owing to the fact that these creeping orbits are much longer than the stationary chords, one can, inter alia, "hear" the stationary chords of Andreev billiards.

Semiclassical analysis of the lowest-order multipole deformations of simple metal clusters

We use a perturbative semiclassical trace formula to calculate the three lowest-order multipole (quadrupole ϵ 2, octupole ϵ 3, and hexadecapole ϵ 4) deformations of simple metal clusters with 90⩽ N⩽550 atoms in their ground states. The self-consistent mean field of the valence electrons is modeled by an axially deformed cavity and the oscillating part of the total energy is calculated semiclassically using the shortest periodic orbits. The average energy is obtained from a liquid-drop model adjusted to the empirical bulk and surface properties of the sodium metal. We obtain good qualitative agreement with the results of quantum-mechanical calculations using Strutinsky's shell-correction method.

Parabolic maps with spin: generic spectral statistics with non-mixing classical limit

We investigate quantized maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i.e. it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantization without altering the classical limit of the dynamics on the torus, we numerically observe a transition to RMT statistics. The results are interpreted in terms of semiclassical trace formulae for the maps with and without spin, respectively. We thus have constructed quantum systems with non-mixing classical limit which show generic (i.e. RMT) spectral statistics. We also discuss the analogous situation for an almost-integrable map, where we compare to semi-Poissonian statistics.

Semiclassical trace formulae using coherent states

AbstractWe derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

Semiclassical trace formulae using coherent states

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

Periodic-orbit bifurcations and superdeformed shell structure.

We have derived a semiclassical trace formula for the level density of the three-dimensional spheroidal cavity. To overcome the divergences occurring at bifurcations and in the spherical limit, the trace integrals over the action-angle variables were performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell-correction energies are in good agreement with quantum-mechanical results. We find that the bifurcations of some dominant short periodic orbits lead to an enhancement of the shell structure for "superdeformed" shapes related to those known from atomic nuclei.

Quantum cat maps with spin ½

We derive a semiclassical trace formula for quantized chaotictransformations of the torus coupled to a 2-spinor precessing in amagnetic field. The trace formula is applied to semiclassical correlationdensities of the quantum map, which, according to the conjecture ofBohigas, Giannoni and Schmit, are expected to converge to those of thecircular symplectic ensemble (CSE) of random matrices. In particular, weshow that the diagonal approximation of the spectral form factor forsmall arguments agrees with the CSE prediction. The results areconfirmed by numerical investigations.

Semiclassical Trace Formulas in Terms of Phase Space Path Integrals

Semiclassical trace formulas are examined using phase space path integrals. Our main concern in this paper is the Maslov index of the periodic orbit, which seems not fully understood in previous works. We show that the calculation of the Maslov index is reduced to a classification of connections on a vector bundle over S1 with structure group Sp(2n, R). We derive a formula for the index of the n-repetition, and show that a Bohr–Sommerfeld type quantization condition including quadratic fluctuation around the orbit is derived using this formula.

Quantum energy levels and classical periodic orbits: discreteness and statistical duality

It is shown that both universal and non-universal correlations must exist between classical periodic orbits in order that Gutzwiller's semiclassical trace formula is consistent with a real, discrete quantum energy spectrum. Formulae for the two-point correlations are derived. The universal correlations are consistent with those conjectured by Argaman et al. (1993). Likewise, both universal and non-universal correlations must exist between quantum energy levels in order that the trace formula be consistent with the fact that periodic orbit actions are real and discrete. In this case, the two-point correlations implied are consistent with random matrix theory and previous semiclassical calculations. These ideas are illustrated with reference to the primes and the Riemann zeros.

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Pade approximant, or signal diagonalization. The second method is based upon the direct application of the Pade approximant to the periodic orbit sum. The Pade approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.

Semiclassical expectation values for relativistic particles with spin 1/2

For relativistic particles with spin 1/2, which are described by the Dirac equation, a semiclassical trace formula is introduced that incorporates expectation values of observables in eigenstates of the Dirac-Hamiltonian. Furthermore, the semiclassical limit of an average of expectation values is expressed in terms of a classical average of the corresponding classical observable.