Fluctuation, time-correlation function and geometric Phase

We establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change to the time integral of the quantum correlation function between the projection operator and force operator of the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalised susceptibility. Relation between the open-path geometric phase, diagonal elements of the quantum metric tensor and the force-force correlation function is provided and the classical limit of the fluctuation-correlation theorem is also discussed.Comment: Latex, 12 pages, no figures, submitted to J. Phys. A: Math & Ge

Quantum ergodicity of C* dynamical systems

This paper contains a very simple and general proof that eigenfunctions of quantizations of classically ergodic systems become uniformly distributed in phase space. This ergodicity property of eigenfunctions f is shown to follow from a convexity inequality for the invariant states (Af,f). This proof of ergodicity of eigenfunctions simplifies previous proofs (due to A.I. Shnirelman, Colin de Verdiere and the author) and extends the result to the much more general framework of C* dynamical systems.Comment: Only very minor differences with the published versio

Dynamic mode II delamination in through thickness reinforced composites

Through thickness reinforcement (TTR) technologies have been shown to provide effective delamination resistance for laminated composite materials. The addition of this reinforcement allows for the design of highly damage tolerant composite structures, specifically when subjected to impact events. The aim of this investigation was to understand the delamination resistance of Z-pinned composites when subjected to increasing strain rates. Z-pinned laminated composites were manufactured and tested using three point end notched flexure (3ENF) specimens subjected to increasing loading rates from quasi-static (~0m/s) to high velocity impact (5m/s), using a range of test equipment including drop weight impact tower and a split Hopkinson bar (SHPB). Using a high speed impact camera and frame by frame pixel tracking of the strain rates, delamination velocities as well as the apparent fracture toughness of the Z-pinned laminates were measured and analysed. Experimental results indicate that there is a transition in the failure morphology of the Z-pinned laminates from quasi-static to high strain rates. The fundamental physical mechanisms that generate this transition are discussed

(Re)constructing Dimensions

Compactifying a higher-dimensional theory defined in R^{1,3+n} on an n-dimensional manifold {\cal M} results in a spectrum of four-dimensional (bosonic) fields with masses m^2_i = \lambda_i, where - \lambda_i are the eigenvalues of the Laplacian on the compact manifold. The question we address in this paper is the inverse: given the masses of the Kaluza-Klein fields in four dimensions, what can we say about the size and shape (i.e. the topology and the metric) of the compact manifold? We present some examples of isospectral manifolds (i.e., different manifolds which give rise to the same Kaluza-Klein mass spectrum). Some of these examples are Ricci-flat, complex and K\"{a}hler and so they are isospectral backgrounds for string theory. Utilizing results from finite spectral geometry, we also discuss the accuracy of reconstructing the properties of the compact manifold (e.g., its dimension, volume, and curvature etc) from measuring the masses of only a finite number of Kaluza-Klein modes.Comment: 23 pages, 3 figures, 2 references adde

L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case

We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $\int_{\gamma} |\phi_{j}^{\hbar}|^2 ds = {\mathcal O}(|\log \hbar|)$ for generic curves $\gamma$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions.Comment: Correct some typos and added some more detail in section

Scarring Effects on Tunneling in Chaotic Double-Well Potentials

The connection between scarring and tunneling in chaotic double-well potentials is studied in detail through the distribution of level splittings. The mean level splitting is found to have oscillations as a function of energy, as expected if scarring plays a role in determining the size of the splittings, and the spacing between peaks is observed to be periodic of period {$2\pi\hbar$} in action. Moreover, the size of the oscillations is directly correlated with the strength of scarring. These results are interpreted within the theoretical framework of Creagh and Whelan. The semiclassical limit and finite-{$\hbar$} effects are discussed, and connections are made with reaction rates and resonance widths in metastable wells.Comment: 22 pages, including 11 figure

Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by $\nu_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-\nu_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.) The main result of the present work is that the nodal deficiency for generic eigenvectors equals to a Morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue. The association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs [arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$ [arXiv:1107.3489]. The present work adapts this result to the discrete case. The definition of the energy functional in the discrete case requires a special setting, substantially different from the one used in [arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.Comment: 15 pages, 1 figur

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 Green function. Semiclassical contributions to the trace show an $\hbar$ - dependent transition from hard chaos to integrable behavior 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 $\hbar$-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.Comment: 27 pages, 6 figures, submitted to J. Phys.