Uzy Smilansky

Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

Abstract We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the periodic orbits proliferate exponentially. An exact trace formula for the quantum spectrum is developed in terms of the same periodic orbits, and it is used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in quantum chaos and related fields.

Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.

Can one hear the shape of a graph

We show that the spectrum of the Schroperator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.

Chaotic Scattering on Graphs

Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.

Quantum graphs: a simple model for chaotic scattering

We connect quantum graphs with infinite leads, and turn them into scattering systems. We show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics: typical poles, delay time and conductance distributions, Ericson fluctuations, and when considered statistically, the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles. The underlying classical dynamics can be defined, and it provides important parameters which are needed for the quantum theory. In particular, we derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. We use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields.

Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization

We study the excitation of molecular rotation by microwave pulses of duration σ which occur periodically with frequency ω. We analyze the molecular dynamics both classically and quantum mechanically and consider situations where the coupling of the field to the molecule is strong. In both approaches, the angular momentum transmitted to the molecule is confined to a finite band of width ≊1/σ. But, while the classical dynamics displays chaotic features, the quantum treatment distinguishes clearly between two regimes. Resonance excitation occurs when ω is rationally related to the basic rotation frequency ω0. Off resonance (ω/ω0 irrational), the probability to transfer angular momentum to the molecule is small and the underlying mechanism for this effect is analogous to the Anderson model of localization in a one‐dimensional random lattice with a finite number of sites. We show that the conditions required by our analysis can be achieved with, e.g., PbTe or CsI molecules and conventional field strengths and ...

A new approach to Gaussian path integrals and the evaluation of the semiclassical propagator

Abstract The expansion of path variations in terms of solutions of Morses boundary problem is applied in order to evaluate Gaussian path integrals. Together with a recently discovered theorem on infinite products of eigenvalues of Sturm-Liouville type operators this yields an expression for the most general semiclassical propagator. The properties of the latter are investigated in the light of the Morse theory. The general methods developed here are illustrated by the example of a charged particle moving in a homogeneous magnetic field.

SEMICLASSICAL QUANTIZATION OF BILLIARDS WITH MIXED BOUNDARY CONDITIONS

The semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived. The parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards. It is also used to check in detail some recently proposed parametric spectral statistics. The methods are illustrated in the analysis of the spectrum of the Sinai billiard and its parametric dependence on the boundary condition on the dispersing arc.

Nodal domains on isospectral quantum graphs: the resolution of isospectrality

We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in which were introduced recently in Gordon et al (1992 Bull. Am. Math. Soc. 27 134–8), Chapman (1995 Am. Math. Mon. 102 124), Buser et al (1994 Int. Math. Res. Not. 9 391–400), Okada and Shudo (2001 J. Phys. A: Math. Gen. 34 5911–22), Jakobson et al (2006 J. Comput. Appl. Math. 194 141–55) and Levitin et al (2006 J. Phys. A: Math. Gen. 39 2073–82)) all based on Sunadas construction of isospectral domains (Sunada T 1985 Ann. Math. 121 196–86). After presenting some of the properties of these graphs, we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.

Fusion of O 16 + Sm 1 4 8 , 1 5 0 , 1 5 2 , 1 5 4 at sub-barrier energies

Measurements of the cross section for fusion of /sup 16/O with /sup 148,150,152,154/Sm have been made in the range 60< or =EO< or =75 MeV. Evaporation residues trapped in a carbon catcher foil were observed off line by means of the K x rays emitted by radioactive Yb nuclei and their daughters. Absolute cross sections varying in magnitude from 0.1 to 400 mb were determined with an uncertainty of +- 10%. The cross sections for individual x-n channels were also determined. At high energies, the fusion cross sections for all isotopes are similar, whereas at lower bombarding energies the cross sections for the more deformed targets are larger than those for the spherical targets.