Gregor Tanner

The semiclassical helium atom

Recent progress in the semiclassical description of two-electron atoms is reported herein. It is shown that the classical dynamics for the helium atom is of mixed phase space structure, i.e., regular and chaotic motion coexists. Semiclassically, both types of motion require separate treatment. Stability islands are quantized via a torus-quantization-type procedure, whereas a periodic-orbit cycle expansion approach accounts for the states associated with hyperbolic electron pair motion. The results are compared with highly accurate ab initio quantum calculations, most of which are reported here for the first time. The results are discussed with an emphasis on previous interpretations of doubly excited electron states

Wave chaos in acoustics and elasticity

Interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations. Forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area, especially in elasticity and acoustics. The theory has indeed been developed primarily in a quantum context; it is increasingly becoming clear, however, that important applications lie in the field of mechanical vibrations and acoustics. We provide an overview over basic concepts in this emerging field of wave chaos. This ranges from ray approximations of the Green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics—both experimentally and from a theoretical/numerical point of view.

Dynamical energy analysis—Determining wave energy distributions in vibro-acoustical structures in the high-frequency regime

We propose a new approach towards determining the distribution of mechanical and acoustic wave energy in complex built-up structures. The technique interpolates between standard Statistical Energy Analysis (SEA) and full ray tracing containing both these methods as limiting case. By writing the flow of ray trajectories in terms of linear phase space operators, it is suggested here to reformulate ray-tracing algorithms in terms of boundary operators containing only short ray segments. SEA can now be identified as a low resolution ray tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. The new technique presented here enhances the range of applicability of standard SEA considerably by systematically incorporating dynamical correlations wherever necessary. Some of the inefficiencies inherent in typical ray tracing methods can be avoided using only a limited amount of the geometrical ray information. The new dynamical theory - Dynamical Energy Analysis (DEA) - thus provides a universal approach towards determining wave energy distributions in complex structures.

Unitary-stochastic matrix ensembles and spectral statistics

We propose to study unitary matrix ensembles defined in terms of unitary-stochastic transition matrices associated with Markov processes on graphs. We argue that the spectral statistics of such an ensemble (after ensemble averaging) depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix. It is conjectured that unitary-stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size N→∞ if the spectral gap of the corresponding transition matrices closes slower than 1/N. The hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non-uniformly connected star graphs and diffusive networks in arbitrary dimensions.

Spectral statistics for unitary transfer matrices of binary graphs

Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalize this approach by considering arbitrary, directed graphs with unitary transfer matrices. An exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes. A special class of graphs, so-called binary graphs, is studied in more detail. For these, the conditions for periodic orbit pairs to be correlated (including correlations due to the unitarity of the transfer matrix) can be given explicitly. Using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low-dimensional cases. Gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs.

HOW CHAOTIC IS THE STADIUM BILLIARD ? A SEMICLASSICAL ANALYSIS

The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing-ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase-space dynamics near the bouncing-ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Greens function. Semiclassical contributions to the trace show an -dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing-ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly -dependent. The localized bouncing-ball states found in the billiard derive from this semiclassically stable island. The bouncing-ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing-ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics. The behaviour is generically found at the border of classically stable islands in systems with a mixed phase-space structure.

A Phase-Space Approach for Propagating Field-Field Correlation Functions

Radiation from complex and inherently random but correlated wave sources can be modelled by exploiting the connection between correlation functions and the Wigner function. Wave propagation can then be directly linked to the evolution of ray densities in phase space. We discuss here in particular the role of evanescent waves in the near-field of non-paraxial sources. We give explicit expressions for the growth rate of the correlation length as function of the distance from the source.

Families of Line-Graphs and Their Quantization

Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family Ln(G). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.

Solving the stationary Liouville equation via a boundary element method

Energy distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, related methods such as Statistical Energy Analysis or variants thereof have found widespread applications. We present a new method for solving the transport equations for complex multi-component structures based on a boundary element formulation of the stationary Liouville equation. The method is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors. It interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.

The semiclassical resonance spectrum of hydrogen in a constant magnetic field

We present the first purely semiclassical calculation of the resonance spectrum in the diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with . The classical system is unbound and completely chaotic for a scaled energy larger than a critical value . The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron - nucleus distance causes divergences in the periodic orbit formula. Using a regularization technique introduced in (Tanner G and Wintgen D 1995 Phys. Rev. Lett. 75 2928) together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi-Einstein - Brillouin - Keller (QEBK) quantization is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg-like series converging towards the different Landau thresholds.