A n a l y s i s o n G r a p h s

a n d I t s A p p l i c a t i o n s

http://dx.doi.org/10.1090/pspum/077

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P r o c e e d i n g s o f S y m p o s i a i n

P U R E M A T H E M A T I C S

V o l u m e 7 7

A n a l y s i s o n G r a p h s

a n d I t s A p p l i c a t i o n s

I s a a c N e w t o n I n s t i t u t e fo r M a t h e m a t i c a l S c i e n c e s ,

C a m b r i d g e , U K

J a n u a r y 8 - J u n e 2 9 , 2 0 0 7

P a v e l E x n e r

J o n a t h a n P . K e a t i n g

P e t e r K u c h m e n t

T o s h i k a z u S u n a d a

A l e x a n d e r T e p l y a e v

E d i t o r s

' TPHTOX A m e r i c a n M a t h e m a t i c a l S o c i e t y Providence, Rhode Island

2000 Mathematics Subject Classification. Primary 05C90, 11M41, 20F65, 28A80, 35-XX, 47-XX, 58-XX, 68R10, 70Q05, 78Axx, 81-XX.

Photo courtesy of Sergey Dobrokhotov.

Library of Congress Cataloging-in-Publication Data Exner, Pavel, 1946-

Analysis on graphs and its applications / Pavel Exner, Jonathan Peter Keating, Peter Kuch-ment.

p. cm. — (Proceedings of symposia in pure mathematics ; v. 77) Includes bibliographical references. ISBN 978-0-8218-4471-7 (alk. paper) 1. Graph theory—Congresses. 2. Quantum graphs—Congresses. 3. Combinatorial analysis—

Congresses. 4. Graphic methods—Congresses. I. Keating, Jonathan Peter, 1963- II. Kuch-ment, Peter, 1949- III. Title.

QA166.E96 2008 511'.5—dc22 2008011370

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Dedicated to the memory of Professor Vladimir Geyler

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C o n t e n t s

Preface xi

Vladimir A. Geyler, April 29, 1943 - April 2, 2007 xiii

C h a p t e r 1. Ana lys i s o n combinator ia l graphs

R e v i e w artic les

Nodal domains on graphs - How to count them and why? RAM BAND, IDAN OREN, AND UZY SMILANSKY 5

Zeta functions of weighted graphs and covering graphs MATTHEW D. HORTON, H. M. STARK, AND AUDREY A. TERRAS 29

Discrete geometric analysis TOSHIKAZU SUNADA 51

Research articles

Uniform existence of the integrated density of states for combinatorial and metric graphs over Zd

MICHAEL J. GRUBER, DANIEL H. LENZ, AND IVAN VESELIC 87

Bartholdi zeta functions for periodic simple graphs DANIELE GUIDO, TOMMASO ISOLA, AND MICHEL L. LAPIDUS 109

Asymptotic properties of Markov processes on Cayley trees M. KELBERT AND Y. SUHOV 123

viii CONTENTS

Chapter 2. Ana lys i s o n fractals

R e v i e w articles

Groups and analysis on fractals VOLODYMYR NEKRASHEVYCH AND ALEXANDER TEPLYAEV 143

Research art ic les

Schreier spectrum of the Hanoi Towers group on three pegs ROSTISLAV GRIGORCHUK AND ZORAN SUNIC 183

On the dichotomy in the heat kernel two sided estimates ALEXANDER GRIGOR'YAN AND TAKASHI KUMAGAI 199

Tube formulas for self-similar fractals MICHEL L. LAPIDUS AND ERIN P. J. PEARSE 211

Existence of eigenforms on nicely separated fractals ROBERTO PEIRONE 231

C h a p t e r 3 . Ana lys i s o n q u a n t u m graphs

R e v i e w artic les

Trace formulae for quantum graphs JENS BOLTE AND SEBASTIAN ENDRES 247

Quantum graphs with spin Hamiltonians J. M. HARRISON 261

Quantum graphs and quantum chaos J. P . KEATING 279

Quantum graphs: an introduction and a brief survey P E T E R KUCHMENT

CONTENTS ix

Research articles

Two constructions of quantum graphs and two types of spectral statistics G. BERKOLAIKO 315

An example on the discrete spectrum of a star graph B. M. BROWN, M. S. P . EASTHAM, AND I. G. W O O D 331

The HELP inequality on trees B. MALCOLM BROWN, MATTHIAS LANGER,

AND KARL MICHAEL SCHMIDT 337

Boundary value problems for infinite metric graphs ROBERT CARLSON 355

Remarks about Hardy inequalities on metric trees TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVARIK 369

On the spectral gap in Andreev graphs HOLGER FLECHSIG AND SVEN GNUTZMANN 381

An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graphs

G. FREILING, M. IGNATIEV, V. YURKO 397

Optimal Wegner estimates for random Schrodinger operators on metric graphs MICHAEL J. GRUBER, MARIO HELM, AND IVAN VESELIC 409

Contraction semigroups on metric graphs VADIM KOSTRYKIN, JURGEN POTTHOFF, AND ROBERT SCHRADER 423

Localization in a quasiperiodic model on quantum graphs KONSTANTIN PANKRASHKIN 459

Equilateral quantum graphs and boundary triples OLAF P O S T 469

A conditionally convergent trace formula for quantum graphs B. W I N N 491

C h a p t e r 4. Appl i ca t ions

R e v i e w artic les

Control problems on quantum graphs SERGEI AVDONIN 507

x CONTENTS

Leaky quantum graphs: a review PAVEL EXNER 523

Thin tubes in mathematical physics, global analysis and spectral geometry DANIEL GRIESER 565

Simulation of quantum graphs by microwave networks OLEH HUL, MICHAL LAWNICZAK, SZYMON BAUCH,

AND LESZEK SIRKO 595

Twisting versus bending in quantum waveguides DAVID KREJCIRIK 617

Research art ic les

Quantum field theory on star graphs B. BELLAZZINI, M. BURRELLO, M. MINTCHEV, AND P. SORBA 639

On the skeleton method and an application to a quantum scissor H.D. CORNEAN, P. DUCLOS, AND B. RlCAUD 657

Vacuum energy and closed orbits in quantum graphs S. A. FULLING AND J. H. WILSON 673

Spectra of graphs and semi-conducting polymers PHILIPP SCHAPOTSCHNIKOW AND SVEN GNUTZMANN 691

P r e f a c e

This volume contains papers written by some of the participants in the program "Analysis on Graphs and its Applications" (AGA) that ran at the Isaac Newton Institute for Mathematical Sciences (INI), Cambridge, from January 8th until June 29th 2007. The organizers of the program were M. Brown (UK), P. Exner (Czech Republic), P. Kuchment (USA), and T. Sunada (Japan).

Let us start by explaining the title of the program (and hence also of this volume). The term "analysis on graphs" was used to cover a variety of problems relating to graphs that have a strongly analytic flavor, either as regards their formu­lation or the techniques used to approach them. Examples include graph analogs of differential equations, spectral theory, differential geometry, and geometric anal­ysis. Problems of this kind have proliferated recently through many branches of Mathematics, the Natural Sciences, and Engineering, including, for instance, num­ber theory, the theory of discrete groups, probability theory, various branches of computer science, optics, quantum mechanics, waveguide theory, nanophysics and nanotechnology, chemistry, microelectronics, materials science, and biological mod­eling (e.g. models of lungs). In many cases, maybe even most, researchers in each area were not aware of the developments, methods and techniques available in oth­ers. The aim of the program was to bring together experts from various disciplines to share and advance their knowledge.

In addition to new problems and techniques, novel graph-like objects have also been introduced. For instance, metric and quantum graphs have been the subject of intense recent interest (see Chapters 3 and 4), and there has been considerable attention paid to analysis on fractals (self-similar structures). Thus the program and this volume naturally split into the following intertwined parts: analysis on graphs in the traditional sense (i.e. on combinatorial graphs), on quantum graphs, and on fractals.

The AGA program at the INI was structured as follows (for further details see http://www.newton.cam.ac.uk/programmes/AGA/index.html). There were around 140 long term participants, each of whom visited the INI for a prolonged period, anywhere from two weeks to six months. These participants engaged in formal dis­cussions at the weekly seminars and, perhaps more importantly, informal meetings in personal offices and at the coffee machine and blackboards distributed through­out the splendid INI building. In addition, there were four formal workshops (which brought the total number of participants to about 200), as detailed below.

A tutorial "Analysis on Graphs and its Applications" (designated as a London Mathematical Society Short Course) was held during the period 1 0 - 1 5 January 2007 as a Satellite Meeting at Gregynog, Newtown, Wales. Most attendees were stu­dents and postdocs. Lecture courses were delivered by T. Sunada (Meiji University,

xi

Xll PREFACE

Japan) on "Spectral geometry of discrete Laplacians," P. Exner (Nuclear Physics Institute, Czech Republic) and P. Kuchment (Texas A & M University, USA) on "Quantum graphs and their applications," A. Teplyaev (University of Connecti­cut, USA) on "Analysis on fractals", and A. Valette (Institut de Mathematiques, Switzerland) on "Ramanujan Graphs". A guest lecture "Spectral statistics" was given by U. Smilansky (Weizmann Institute of Science, Israel).

A research workshop on "Quantum Graphs, their Spectra and Applications" took place at the INI from 2nd to 5th of April 2007. A follow up workshop "Graph Models of Mesoscopic Systems, Wave-Guides and Nano-Structures" at the INI from 10th to 13th of April mostly concentrated on applications.

The final workshop "Analysis on Graphs and Fractals", joint with a meeting of the London Mathematical Society, took place between 29th of May and 2nd of June as a Satellite Meeting at Cardiff University.

Long term visitors and participants of the workshops came from many branches of mathematics, as well as from physics, computer sciences, and chemistry.

The program was, we believe, extremely successful in terms of sharing tech­niques, fostering new collaborations and stimulating progress in this fast developing interdisciplinary field.

The way this volume is arranged reflects to a large extent the structure of the program. The first chapter is devoted to analysis on truly discrete structures: combinatorial graphs (i.e. graphs where the edges are treated as relations between vertices rather than one-dimensional cells). The next chapter addresses analysis on quantum graphs, i.e. graphs considered as singular one-dimensional manifolds and equipped with differential (or sometimes pseudo-differential) operators. Chapter 3 focuses on analysis on fractals. Applications are considered in the final chapter (although some applications appear in the previous chapters as well).

Each chapter contains a selection of review articles, followed by more focused research papers. The distinction is, however, a matter of judgement, since many papers include both a survey and new results.

Generous support for the program at the INI was provided by various orga­nizations, including the UK Engineering and Physical Sciences Research Council (EPSRC), the European program SPECT, the London Mathematical Society, and the US National Science Foundation (NSF). The organizers and the editors express their gratitude for this support. The friendly and magnificently efficient staff of the Isaac Institute for Mathematical Sciences, and in particular its Director Sir David Wallace, have created a wonderful environment, for which we are also extremely grateful. We also thank all participants of the program and especially all contribu­tors to this volume. Finally, we are grateful to the American Mathematical Society staff members who have made publication of this volume possible, and in particular Dr. S. Gelfand and Ms. C. M. Thivierge for their tireless work with the editors and the authors.

Unfortunately, we have to finish on a solemn note. On the first day of the workshop on Quantum Graphs, one of the leading participants in the program, Professor Vladimir Geyler, passed away. We dedicate this volume to his memory.

P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev

V l a d i m i r A . G e y l e r , A p r i l 2 9 , 1 9 4 3 - A p r i l 2 , 2 0 0 7

Professor Vladimir Geyler, a prominent mathematical physicist and one of the leading researchers in the field of Analysis on Graphs, passed away unexpectedly on April 2nd 2007 in Cambridge. He was a highly esteemed colleague and co-author of many of the participants of the program at the Isaac Newton Institute, and he played an important role in the development of a number of the topics addressed in this volume. He was also an exceptionally sensitive and generous man who was held in the highest affection by his many friends. We devote this volume to his memory, which we cherish.

A biography of Professor Geyler, together with a complete list of his publica­tions, can be found in a memorial issue (number 4, volume 14, 2007) of the Russian Journal of Mathematical Physics.

P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev

x i i i

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