Semiclassical Approach to Parametric Spectral Correlation with Spin 1/2

The spectral correlation of a chaotic system with spin 1/2 is universally described by the GSE (Gaussian Symplectic Ensemble) of random matrices in the semiclassical limit. In semiclassical theory, the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere. In this paper, instead of the uniform distribution, we introduce Brownian motion on a sphere to yield the parametric motion of the energy levels. As a result, the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the GSE universality class. Moreover, by starting the Brownian motion from a point distribution on the sphere, we gradually increase the effect of the spin and calculate the form factor describing the transition from the GOE (Gaussian Orthogonal Ensemble) class to the GSE class.Comment: 25 pages, 2 figure

Spectral Statistics for the Dirac Operator on Graphs

We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantisation condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

Quantum graphs with singular two-particle interactions

We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions. In order to find such Hamiltonians closed and semi-bounded quadratic forms are constructed, from which the associated self-adjoint operators are extracted. We provide a general characterisation of such operators and, furthermore, produce certain classes of examples. We then consider identical particles and project to the bosonic and fermionic subspaces. Finally, we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate Weyl asymptotic law

Trace Formulae for quantum graphs with edge potentials

This work explores the spectra of quantum graphs where the Schr\"odinger operator on the edges is equipped with a potential. The scattering approach, which was originally introduced for the potential free case, is extended to this case and used to derive a secular function whose zeros coincide with the eigenvalue spectrum. Exact trace formulas for both smooth and $\delta$-potentials are derived, and an asymptotic semiclassical trace formula (for smooth potentials) is presented and discussed

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

Semiclassical Approach to Chaotic Quantum Transport

We describe a semiclassical method to calculate universal transport properties of chaotic cavities. While the energy-averaged conductance turns out governed by pairs of entrance-to-exit trajectories, the conductance variance, shot noise and other related quantities require trajectory quadruplets; simple diagrammatic rules allow to find the contributions of these pairs and quadruplets. Both pure symmetry classes and the crossover due to an external magnetic field are considered.Comment: 33 pages, 11 figures (appendices B-D not included in journal version

Extented ionized gas emission and kinematics of the compact group galaxies in HCG 16: Signatures of mergers

We report on kinematic observations of Ha emission line from four late-type galaxies of Hickson Compact Group 16 (H16a,b,c and d) obtained with a scanning Fabry-Perot interferometer and samplings of 16 km/s and 1". The velocity fields show kinematic peculiarities for three of the four galaxies: H16b, c and d. Misalignments between the kinematic and photometric axes of gas and stellar components (H16b,c,d), double gas systems (H16c) and severe warping of the kinematic major axis (H16b and c) were some of the peculiarities detected. We conclude that major merger events have taken place in at least two of the galaxies group. H16c and d, based on their significant kinematic peculiarities, their double nuclei and high infrared luminosities. Their Ha gas content is strongly spatially concentred - H16d contains a peculiar bar-like structure confined to the inner $\sim$ 1 h^-1 kpc region. These observations are in agreement with predictions of simulations, namely that the gas flows towards the galaxy nucleus during mergers, forms bars and fuel the central activity. Galaxy H16b, and Sb galaxy, also presents some of the kinematic evidences for past accretion events. Its gas content, however, is very spare, limiting our ability to find other kinematic merging indicators, if they are present. We find that isolated mergers, i.e., they show an anormorphous morphology and no signs of tidal tails. Tidal arms and tails formed during the mergers may have been stripped by the group potential (Barnes & Hernquist 1992) ar alternatively they may have never been formed. Our observations suggest that HCG 16 may be a young compact group in formation throught the merging of close-by objects in a dense environment.Comment: Accepted for publication in ApJ. 35 pages, 13 figures. tar file gzipped and uuencode

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

Two-point correlations of the Gaussian symplectic ensemble from periodic orbits

We determine the asymptotics of the two-point correlation function for quantum systems with half-integer spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the leading terms of the two-point correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the role of Kramers' degeneracy.Comment: 7 pages, no figure

Zero Modes of Quantum Graph Laplacians and an Index Theorem

We study zero modes of Laplacians on compact and non-compact metric graphs with general self-adjoint vertex conditions. In the first part of the paper the number of zero modes is expressed in terms of the trace of a unitary matrix $\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the domain of the Laplacian. In the second part a Dirac operator is defined whose square is related to the Laplacian. In order to accommodate Laplacians with negative eigenvalues it is necessary to define the Dirac operator on a suitable Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph Laplacian admits a factorisation into momentum-like operators in a Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for the associated Dirac operator and prove that the zero-mode contribution in the trace formula for the Laplacian can be expressed in terms of the index of the Dirac operator