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Book DetailsContemporary MathematicsVolume: 415; 2006; 307 ppMSC: Primary 05; 11; 15; 28; 34; 35; 58; 60; 81;
This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.
ReadershipGraduate students and research mathematicians interested in quantum graphs and application to number theory, mathematical physics, and other areas.

Table of Contents

Articles

Michael Aizenman, Robert Sims and Simone Warzel  Fluctuationbased proof of the stability of ac spectra of random operators on tree graphs [ MR 2277604 ]

Matthew Baker and Xander Faber  Metrized graphs, Laplacian operators, and electrical networks [ MR 2277605 ]

Gregory Berkolaiko  Form factor expansion for large graphs: a diagrammatic approach [ MR 2277606 ]

Jens Bolte and Jonathan Harrison  The spectral form factor for quantum graphs with spinorbit coupling [ MR 2277607 ]

Robert Carlson  Linear network models related to blood flow [ MR 2277608 ]

Kingwood Chen, Stanislav Molchanov and Boris Vainberg  Localization on AvronExnerLast graphs. I. Local perturbations [ MR 2277609 ]

Fan Chung and Ross M. Richardson  Weighted Laplacians and the sigma function of a graph [ MR 2277610 ]

Pavel Exner and Ondřej Turek  Approximations of permutationsymmetric vertex couplings in quantum graphs [ MR 2277611 ]

Daniel Fontaine, Thomas Smith and Alexander Teplyaev  Resistance of random Sierpiński gaskets [ MR 2277612 ]

Mark Freidlin and Matthias Weber  Small diffusion asymptotics for exit problems on graphs [ MR 2277613 ]

Leonid Friedlander  Determinant of the Schrödinger operator on a metric graph [ MR 2277614 ]

Stephen A. Fulling  Local spectral density and vacuum energy near a quantum graph vertex [ MR 2277615 ]

Matthew D. Horton, H. M. Stark and Audrey A. Terras  What are zeta functions of graphs and what are they good for? [ MR 2277616 ]

J. P. Keating  Fluctuation statistics for quantum star graphs [ MR 2277617 ]

Vadim Kostrykin and Robert Schrader  Laplacians on metric graphs: eigenvalues, resolvents and semigroups [ MR 2277618 ]

Stanislav Molchanov and Boris Vainberg  Transition from a network of thin fibers to the quantum graph: an explicitly solvable model [ MR 2277619 ]

BengSeong Ong  On the limiting absorption principle and spectra of quantum graphs [ MR 2277620 ]

Jacob Rubinstein  Quantum mechanics, superconductivity and fluid flow in narrow networks [ MR 2277621 ]

Holger Schanz  A relation between the bond scattering matrix and the spectral counting function for quantum graphs [ MR 2277622 ]

Uzy Smilansky and Michael Solomyak  The quantum graph as a limit of a network of physical wires [ MR 2277623 ]

B. Winn  On the trace formula for quantum star graphs [ MR 2277624 ]


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This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.
Graduate students and research mathematicians interested in quantum graphs and application to number theory, mathematical physics, and other areas.

Articles

Michael Aizenman, Robert Sims and Simone Warzel  Fluctuationbased proof of the stability of ac spectra of random operators on tree graphs [ MR 2277604 ]

Matthew Baker and Xander Faber  Metrized graphs, Laplacian operators, and electrical networks [ MR 2277605 ]

Gregory Berkolaiko  Form factor expansion for large graphs: a diagrammatic approach [ MR 2277606 ]

Jens Bolte and Jonathan Harrison  The spectral form factor for quantum graphs with spinorbit coupling [ MR 2277607 ]

Robert Carlson  Linear network models related to blood flow [ MR 2277608 ]

Kingwood Chen, Stanislav Molchanov and Boris Vainberg  Localization on AvronExnerLast graphs. I. Local perturbations [ MR 2277609 ]

Fan Chung and Ross M. Richardson  Weighted Laplacians and the sigma function of a graph [ MR 2277610 ]

Pavel Exner and Ondřej Turek  Approximations of permutationsymmetric vertex couplings in quantum graphs [ MR 2277611 ]

Daniel Fontaine, Thomas Smith and Alexander Teplyaev  Resistance of random Sierpiński gaskets [ MR 2277612 ]

Mark Freidlin and Matthias Weber  Small diffusion asymptotics for exit problems on graphs [ MR 2277613 ]

Leonid Friedlander  Determinant of the Schrödinger operator on a metric graph [ MR 2277614 ]

Stephen A. Fulling  Local spectral density and vacuum energy near a quantum graph vertex [ MR 2277615 ]

Matthew D. Horton, H. M. Stark and Audrey A. Terras  What are zeta functions of graphs and what are they good for? [ MR 2277616 ]

J. P. Keating  Fluctuation statistics for quantum star graphs [ MR 2277617 ]

Vadim Kostrykin and Robert Schrader  Laplacians on metric graphs: eigenvalues, resolvents and semigroups [ MR 2277618 ]

Stanislav Molchanov and Boris Vainberg  Transition from a network of thin fibers to the quantum graph: an explicitly solvable model [ MR 2277619 ]

BengSeong Ong  On the limiting absorption principle and spectra of quantum graphs [ MR 2277620 ]

Jacob Rubinstein  Quantum mechanics, superconductivity and fluid flow in narrow networks [ MR 2277621 ]

Holger Schanz  A relation between the bond scattering matrix and the spectral counting function for quantum graphs [ MR 2277622 ]

Uzy Smilansky and Michael Solomyak  The quantum graph as a limit of a network of physical wires [ MR 2277623 ]

B. Winn  On the trace formula for quantum star graphs [ MR 2277624 ]