Quantization conditions in Bogomolny's transfer operator method.

This work demonstrates the application of Bogomolny's transfer operator method on three-dimensional dynamics. Motivated by experimental observations of lenslike metal clusters, the quantum billiards bounded by a flat bottom and an upper surface with SO(2) symmetry are studied. A precise determination of the energies with error less than 0.05% and exact predicted degeneracies in the special case of the half-sphere billiard confirm the efficiency of this method. Furthermore, the spectra and degeneracies of lens billiards with varying heights are explicitly determined.

Path integration over closed loops and Gutzwiller's trace formula

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 Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many non-trivial zeros on the real axis. In a special case we can show the existence of also infinitely many trivial zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function.

Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

Transfer operators and Hankel transforms between relative trace formulas, I: Character theory

(1) What is semiclassical Analysis? (2) What is a Trace Formula? (3) Physical Motivations : “Quantum Chaos” (4) Gutzwiller Trace Formula. (Rigorous Approach)

Geometry and non-adiabatic response in quantum and classical systems

The aim of this paper is the construction of numerical tools forthe efficient approximation of transport phenomena in non-autonomous dynamical systems.We focus on transfer operator methods which have been developed in the last yearsfor the treatment of non-autonomous dynamical systems.For instance Froyland et al. [11] proposed a methodfor the approximation of so-called coherent pairs -- these pairs of setsrepresent time-dependent slowly mixing structures -- bythresholding singular vectors of a normalized transfer operator over a fixed time-interval.In principle such transfer operator methods involve long term simulationsof trajectories on the whole state space. In our main result we showthat transport phenomena over a fixed (long) time horizonimply the existence of almost invariant sets over shorter time intervalsif the transport process is slow enough.This fact is used to formulate an algorithm that preselects part of state spaceas a candidate for containing one of the sets of a coherent pair.By this we significantly reduce the related numerical effort.

Quantum chaotic transport in double barrier tunnel structures

The problem of emergence in physical theories makes necessary to build a\ngeneral theory of the relationships between the observed system and the\nobserving system. It can be shown that there exists a correspondence between\nclassical systems and computational dynamics according to the Shannon-Turing\nmodel. A classical system is an informational closed system with respect to the\nobserver; this characterizes the emergent processes in classical physics as\nphenomenological emergence. In quantum systems, the analysis based on the\ncomputation theory fails. It is here shown that a quantum system is an\ninformational open system with respect to the observer and able to exhibit\nprocesses of observational, radical emergence. Finally, we take into\nconsideration the role of computation in describing the physical world.\n

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce the thermodynamic and structural properties. The objective is to develop a means for exploiting strong coupling classical methods (e.g., MD, integral equations, DFT) to describe quantum systems. The classical system has an effective temperature, local chemical potential, and pair interaction that are defined by requiring equivalence of the grand potential and its functional derivatives with respect to the external and pair potentials for the classical and quantum systems. Practical inversion of this mapping for the classical properties is effected via the hypernetted chain approximation, leading to representations as functionals of the quantum pair correlation function. As an illustration, the parameters of the classical system are determined approximately such that ideal gas and weak coupling RPA limits are preserved (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)