Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory

In this paper we present a general framework for solving the stationary nonlinear Schr\"odinger equation (NLSE) on a network of one-dimensional wires modelled by a metric graph with suitable matching conditions at the vertices. A formal solution is given that expresses the wave function and its derivative at one end of an edge (wire) nonlinearly in terms of the values at the other end. For the cubic NLSE this nonlinear transfer operation can be expressed explicitly in terms of Jacobi elliptic functions. Its application reduces the problem of solving the corresponding set of coupled ordinary nonlinear differential equations to a finite set of nonlinear algebraic equations. For sufficiently small amplitudes we use canonical perturbation theory which makes it possible to extract the leading nonlinear corrections over large distances.Comment: 26 page

Quantum Graphs via Exercises

Studying the spectral theory of Schroedinger operator on metric graphs (also known as quantum graphs) is advantageous on its own as well as to demonstrate key concepts of general spectral theory. There are some excellent references for this study such as a mathematically oriented book by Berkolaiko and Kuchment, a review with applications to theoretical physicsby Gnutzmann and Smilansky, and elementary lecture notes by Berkolaiko. Here, we provide a set of questions and exercises which can accompany the reading of these references or an elementary course on quantum graphs. The exercises are taken from courses on quantum graphs which were taught by the authors

Eigenfunction Statistics on Quantum Graphs

We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for which such a model was proposed by Berry in 1977. The autocorrelation functions we calculate for an individual quantum graph exhibit a universal component, which completely determines a Gaussian Random Wave Model, and a system-dependent deviation. This deviation depends on the graph only through its underlying classical dynamics. Classical criteria for quantum universality to be met asymptotically in the large graph limit (i.e. for the non-universal deviation to vanish) are then extracted. We use an exact field theoretic expression in terms of a variant of a supersymmetric sigma model. A saddle-point analysis of this expression leads to the estimates. In particular, intensity correlations are used to discuss the possible equidistribution of the energy eigenfunctions in the large graph limit. When equidistribution is asymptotically realized, our theory predicts a rate of convergence that is a significant refinement of previous estimates. The universal and system-dependent components of intensity correlation functions are recovered by means of an exact trace formula which we analyse in the diagonal approximation, drawing in this way a parallel between the field theory and semiclassics. Our results provide the first instance where an asymptotic Gaussian Random Wave Model has been established microscopically for eigenfunctions in a system with no disorder.Comment: 59 pages, 3 figure

The impact of trade preferences removal: Evidence from the Belarus Generalized System of Preferences withdrawal

Under the Generalized System of Preferences (GSP), high-income countries grant unilateral trade preferences to developing countries. These preferences are subject to political conditionality, but little is known about the trade impact of loss of preferential access. We study the EU's complete withdrawal of GSP preferences from Belarus in 2007 in response to labour rights violations to fill this void. The withdrawal caused a significant drop in trade for affected products (25%–27% trade decline) and some trade reduction at the extensive margin. For products where trade was affected at the intensive margin, there is some evidence of adjustment through falls in quantities but also through prices for larger export sectors. The impact was uneven across sectors, with textiles and plastics particularly strongly affected by the withdrawal

Topological Resonances in Scattering on Networks (Graphs)

We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph - its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances `topological' to emphasize their origin in the non-trivial connectivity. Topological resonances have a clear and unique signature which is apparent in the statistics of the resonance parameters (such as e.g., the width, the delay time or the wave-function intensity in the graph). We discuss this phenomenon by providing analytical arguments supported by numerical simulation, and identify the features of the above distributions which depend on genuine topological quantities such as the length of the shortest cycle (girth). These signatures cannot be explained using any of the other paradigms for narrow resonances. Finally, we propose an experimental setting where the topological resonances could be demonstrated, and study the stability of the relevant distribution functions to moderate dissipation

Resolving isospectral "drums" by counting nodal domains

Several types of systems were put forward during the past decades to show that there exist {\it isospectral} systems which are {\it metrically} different. One important class consists of Laplace Beltrami operators for pairs of flat tori in $\mathbb{R}^n$ with $n\geq 4$. We propose that the spectral ambiguity can be resolved by comparing the nodal sequences (the numbers of nodal domains of eigenfunctions, arranged by increasing eigenvalues). In the case of isospectral flat tori in four dimensions - where a 4-parameters family of isospectral pairs is known- we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count. Thus - one can {\it count} the shape of a drum (if it is designed as a flat torus in four dimensions...).Comment: 13 pages, 3 figure