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

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

Spectra of graphs and semi-conducting polymers

We study the band gap in some semi-conducting polymers with two models: H\"uckel molecular orbital theory and the so-called free electron model. The two models are directly related to spectral theory on combinatorial and metric graphs.Comment: Revised versio

El juguete rabioso: del aprendizaje a la escritura

En el presente trabajo se realiza una lectura de El juguete rabioso (1926) a partir de los diversos encuadres genéricos reconocidos por la crítica. El estudio se apoya en los aportes que reconocen en la obra, la narración de la historia de su protagonista, Silvio Astier, y que relacionan esta narración con la picaresca y el Bildungsroman. El análisis se centra en dos aspectos fundamentales: el aprendizaje de Silvio como actor de su propia historia / ladrón, y el de su aprendizaje como escritor que explica mediante la puesta en texto del relato, la clave de su existencia.Fil: Gnutzmann, Rita

Remarks on "Resolving isospectral `drums' by counting nodal domains"

In [3] the authors studied the 4-parameter family of isospectral flat 4-tori T^\pm(a,b,c,d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a,b,c,d). In this note we confirm the expectation expressed in [3] by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.Comment: 5 page

Sigma models for quantum chaotic dynamics

We review the construction of the supersymmetric sigma model for unitary maps, using the color- flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization

Mitologia y realidad socio-histórica en El Hablador de Vargas Llosa.

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