Accuracy of trace formulas

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.

This chapter is dedicated to the derivation of semiclassical trace formulae for particles with spin. We will focus on Pauli Hamiltonians with arbitrary spin and only briefly describe the corresponding calculations for the Dirac equation. We will also comment on a different limit in which the spectral asymptotics of the Pauli Hamiltonian can be considered.KeywordsPeriodic OrbitDirac EquationJacobi EquationTrace FormulaOrbit CouplingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The principal families of periodic orbits that emerge from the stationary points of the six-dimensional potential energy surface of the C2H2 molecular system, as well as periodic orbits from saddle-node bifurcations, have been located and propagated for an energy range up to 36 500 cm−1 above the absolute minimum of the potential. The bifurcation diagrams of these periodic orbits reveal the regions of phase space where the dynamics are regular or chaotic (with soft or hard chaos) for acetylene, vinylidene, and the region over these two isomers. An association of the structure of phase space with spectroscopic findings is made by calculating Gutzwiller’s semiclassical trace formula and classical survival probability functions.

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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.

The electronic supershell structure of simple metal clusters in a uniform magnetic field is studied with a spherical infinite-well potential as a simple yet realistic model for confining the valence electrons. It is found that there is little perceivable change in the supershells with experimentally achievable field strengths. If yet stronger fields are applied, the supershell structure gets destroyed and then new beat structures come about as the field strength increases further. These phenomena are explained by a recently derived semiclassical trace formula for broken symmetry that has been applied to the present system. \textcopyright{} 1996 The American Physical Society.

We consider the effect of a continuous family of neutral (bouncing ball) orbits on the energy spectrum of the quantized stadium billiard. Using a semiclassical approximation we derive analytic expressions for standard two-point spectral measures. The corrections due to the bouncing ball orbits account for some of the non-generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured. Once the bouncing ball contributions are subtracted, the spectrum is shown to be well reproduced by the semi-classical trace formula based on unstable periodic orbits. We also study special patterns in the spectrum which are due to other non-generic features such as edge effects and 'whispering gallery' trajectories.

A semiclassical approach to the universal ergodic spectral statistics in quantum star graphs is presented for all known ten symmetry classes of quantum systems. The approach is based on periodic orbit theory, the exact semiclassical trace formula for star graphs, and on diagrammatic techniques. The appropriate spectral form factors are calculated up to one order beyond the diagonal and self-dual approximations. The results are in accordance with the corresponding random-matrix theories which supports a properly generalized Bohigas-Giannoni-Schmit conjecture.

In this chapter we shall present the results of a detailed analysis where we have tested both theoretically and experimentally the validity of equation (9.2). We obtain good agreement between the experimental and calculated SdH oscillations. On the other hand, the calculated B = 0 spin splitting differs substantially from the predictions of (9.2). Using a semiclassical trace formula for particles with spin, we shall show that the anomalous magneto-oscillations reflect the nonadiabatic spin precession along the cyclotron orbits.

The leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wave number dependence determines the normal transport coefficients. In particular, a general exact formula for the diffusion coefficient D is derived without any high stochasticity approximation, and a new effect emerges: The angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The behavior of D is examined for three particular cases: (i) the standard map, (ii) a sawtooth map, and (iii) a Harper map as an example of a map with a nonlinear rotation number. Numerical simulations support this formula.

The Frenkel-Kontorova (FK) model describes a chain of atoms connected by springs subject to an external potential. This simple classical model exhibits a wealth of complex behavior. It has also found applications in many condensed matter systems such as charge density waves, magnetic spirals, modulated phases and tribology. However, an in-depth understanding of some of these problems, for example, tribology in the nano-regime, demands an understanding of its quantum mechanical behavior. To achieve this goal, we use a squeezed-state approach rst used in quantum optics. We found that quantum uctuations renormalize the standard map, which governs the classical behavior of the FK model, to a sawtooth map. This result is borne out by Monte-Carlo simulations. We also found that the ground state wave function changes from an extended state to a localized state when the coupling constant increases. Although quantum uctuations largely smear the transition by breaking of analyticity observed in the classical case, the remnant of this transition is still discernible.