Phase-Space Semiclassical Analysis. Around Semiclassical Trace Formulae

Path integration over closed loops and Gutzwiller's trace formula

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of “Gutzwiller's trace formula” as a leading term in an asymptotic expansion around classical periodic orbits.

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.

We investigate the resonance spectrum of the Hénon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller's trace formula, complemented by a uniform approximation for a co-dimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula for single-diffractive orbits which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.

The simplest chaotic Hamiltonian systems arise in geometry: a point particle moving freely on a 2 or 3 dimensional manifold of constant negative curvature. There is tremendous variety representing both bound state and scattering situations, quite in contrast to the non-chaotic motion on surfaces of constant positive curvature. The classical mechanics has been studied for almost a century and reduces to an enumeration and description of geodesic lines which in turn are closely connected with the topology of the manifold. Physics enters only in a trivial manner since everything can be scaled with the (kinetic) energy of the particle. In the quantum mechanics of these motions, however, the energy is the essential parameter, and one can ask how the two mechanics are related to each other. Quite surprisingly, Feynman's path integral gives the same answer as its classical approximation, just as in the harmonic oscillator. In 3 dimensions, this result holds for the propagator itself, while in 2 dimensions it is true only for the trace and yields Selberg's trace formula. A few striking examples of constant negative curvature spaces will be shown. Thus, quantum chaos can be understood in terms of classical chaos in spite of their qualitatively different features: fractal for the latter, smooth for the former.

The scattering of a point particle from three hard discs in a plane is studied in the semiclassical approximation, using the Gutzwiller trace formula. Using a previously introduced coding of the classical dynamics, the needed summation over the classical periodic orbits is performed. The trace function is then given in terms of Ruelle zeta functions. A semiclassical limit upper bound is obtained on the lifetimes of the scattering resonances. This bound is larger than the classical lifetime when the classical repellor is chaotic but coincides with it when the repellor is periodic. We conclude that classical chaos dramatically influences the lifetimes of the scattering resonances. Our upper bound for the resonance lifetime is compared with the results of numerical calculation of the full quantum dynamics. The distribution of the imaginary parts of the complex wave numbers of the resonances is also calculated.

Quantum chaotic transport in double barrier tunnel structures

There are a number of excellent texts available on semi-classical analysis. The focus of the present monograph, however, is an aspect of the subject somewhat less systematically developed in other texts: In semi-classical analysis, many of the basic results involve asymptotic expansions in which the terms can be computed by symbolic techniques, and the focus of this monograph is the symbol created thus. In particular, the techniques involved in this symbolic calculus have their origins in symplectic geometry, and the first seven chapters of the present work are a discussion of this underlying symplectic geometry. Another feature which differentiates this monograph from other texts is an emphasis on the global aspects of the subject: A considerable amount of time is spent here showing that the objects studied are coordinate-invariant and hence make sense on manifolds. Wherever possible, intrinsic coordinate-free descriptions of these objects are given. Topics discussed include wave and heat trace formulas for globally defined semi-classical differential operators on manifolds, and equivariant versions of these results involving Lie group actions.

The leaky tori model for scattering theory in mesoscopic systems was introduced by Gutzwiller in 1983. Muller’s class of admissable surfaces subsumes the leaky tori models of Gutzwiller. In addition, they form a natural generalization of locally symmetric spaces of constant negative curvature and finite volume. In this chapter we review Muller’s spaces in Section 1. In Section 2 scattering operators are developed. In Section 3 Weyl’s law for mesoscopic systems is discussed. In Section 4 Miiller’s trace formula is presented. In Sections 5 to 8 work of Guillope and Zworski (1994) on scattering theory on hyperbolic half-cylinders is developed. Their work uses some classical results of Poschl and Teller which allows one to explicitly deduce the resonance set for the Laplacian on the hyperbolic half-cylinder. In Section 9 we briefly review related results on scattering theory for two strictly convex bodies. And in Section 10 we return to the Gutzwiller trace formula and look at including diffraction terms.

We consider the eigenvalues of a one-dimensional semiclassical Schrödinger operator, where the potential consists of two quadratic ends (that is, looks like a harmonic oscillator at each infinite end), possibly with a flat region in the middle. Such a potential notably has a discontinuity in the second derivative. We derive an asymptotic expansion, valid either in the high energy regime or the semiclassical regime, with a leading order term given by the Bohr–Sommerfeld quantization condition, and an asymptotic expansion consisting of negative powers of the leading order term, with coefficients that are oscillatory in the leading order term. We apply this expansion to study the results of the Gutzwiller trace formula and the heat kernel asymptotic for this class of potentials, giving an idea into what results to expect for such trace formulas for non-smooth potentials.

An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and which correspond to creping waves. A new family of resonances in the two disk scattering system can be well described which is completely missing if only the traditional periodic orbits are used.

We derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the Schr\"odinger equation. The system we considered is an energy-conserved Hamiltonian system possessing time-invariant symmetry. The energy smoothed autocorrelation function is expressed as a sum of three terms. The first one is analogous to Berry's conjecture, which is a Bessel function of the zeroth order. The second and the third terms are trace formulae made from special trajectories. The second term is found to be direction dependent in the case of spacing averaging, which agrees qualitatively with previous numerical observations in high-lying eigenstates of a chaotic billiard.

We present here a review of two problems motivated by 2D models for high T, superconductivity. The first part concerns the energy spectrum of 2D Bloch electrons in a uniform magnetic field. A semiclassical analysis provides a qualitative as well as a quantitative understanding of this spectrum. In the second part we make the case for the application of “Quantum Chaos" to strongly correlated fermion systems. It is illustrated by the level spacing distribution for the t - J model in two dimensions.