Introduction to Quantum Graphs

Mathematical Surveys and Monographs Volume 186

'REGORY"ERKOLAIKO Peter Kuchment

American Mathematical Society http://dx.doi.org/10.1090/surv/186

Introduction to Quantum Graphs

Mathematical Surveys and Monographs Volume 186

Introduction to Quantum Graphs

Gregory Berkolaiko Peter Kuchment

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein

2010 Mathematics Subject Classiﬁcation. Primary 34B45, 35R02, 81Q35, 35Pxx, 58J50, 05C50, 82D77, 81Q50, 47N60, 82D80, 82D55.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-186

Library of Congress Cataloging-in-Publication Data Berkolaiko, Gregory, 1976– Introduction to quantum graphs / Gregory Berkolaiko, Peter Kuchment. pages cm. – (Mathematical surveys and monographs ; volume 186) Includes bibliographical references and index. ISBN 978-0-8218-9211-4 (alk. paper) 1. Quantum graphs. 2. Boundary value problems. I. Kuchment, Peter, 1949– II. Title.

QA166.165.B47 2012 530.12015115–dc23 2012033434

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Contents

Preface xi

Introduction xiii Chapter 1. Operators on Graphs. Quantum graphs 1 1.1. Main graph notions and notation 2 1.2. Difference operators. Discrete Laplace operators 5 1.3. Metric graphs 7 1.4. Differential operators on metric graphs. Quantum graphs 12 1.4.1. Vertex conditions. Finite graphs. 15 1.4.2. Scale invariance 22 1.4.3. Quadratic form 22 1.4.4. Examples of vertex conditions 24 1.4.5. Infinite graphs 27 1.4.6. Non-local vertex conditions 32 1.5. Further remarks and references 33

Chapter 2. Quantum Graph Operators. Special Topics 37 2.1. Quantum graphs and scattering matrices 37 2.1.1. Scattering on vertices 37 2.1.2. Bond scattering matrix and the secular equation 41 2.2. First order operators and scattering matrices 44 2.3. Factorization of quantum graph Hamiltonians 51 2.4. Index of quantum graph operators 52 2.5. Dependence on vertex conditions 54 2.5.1. Variations in the edge lengths 58 2.6. Magnetic Schr¨odinger operator 59 2.7. Further remarks and references 62

Chapter 3. Spectra of Quantum Graphs 65 3.1. Basic spectral properties of compact quantum graphs 66 3.1.1. Discreteness of the spectrum 66 3.1.2. Dependence on the vertex conditions 67 3.1.3. Eigenfunction dependence 68 3.1.4. An Hadamard-type formula 68 vii viii CONTENTS

3.1.5. Generic simplicity of the spectrum 71 3.1.6. Eigenvalue bracketing 72 3.1.7. Dependence on the coupling constant at a vertex 76 3.2. The Shnol’ theorem 79 3.3. Generalized eigenfunctions 82 3.4. Failure of the unique continuation property. Scars 84 3.5. The ubiquitous Dirichlet-to-Neumann map 85 3.5.1. DtN map for a single edge 85 3.5.2. DtN map for a compact graph with a “boundary” 87 3.5.3. DtN map for a single vertex boundary 89 3.5.4. DtN map and the secular equation 89 3.5.5. DtN map and number of negative eigenvalues 90 3.6. Relations between quantum and discrete graph spectra 90 3.7. Trace formulas 92 3.7.1. Secular equation 93 3.7.2. Weyl’s law 95 3.7.3. Derivation of the trace formula 96 3.7.4. Expansion in terms of periodic orbits 99 3.7.5. Other formulations of the trace formula 100 3.8. Further remarks and references 101

Chapter 4. Spectra of Periodic Graphs 105 4.1. Periodic graphs 105 4.2. Floquet-Bloch theory 107 4.2.1. Floquet transform on combinatorial periodic graphs 109 4.2.2. Floquet transform of periodic diﬀerence operators 112 4.2.3. Floquet transform on quantum periodic graphs 113 4.2.4. Floquet transform of periodic operators 114 4.3. Band-gap structure of spectrum 115 4.3.1. Discrete case 115 4.3.2. Quantum graph case 116 4.3.3. Floquet transform in Sobolev classes 117 4.4. Absence of the singular continuous spectrum 117 4.5. The point spectrum 118 4.6. Where do the spectral edges occur? 121 4.7. Existence and location of spectral gaps 123 4.8. Impurity spectra 124 4.9. Further remarks and references 124

Chapter 5. Spectra of Quantum Graphs. Special Topics 129 5.1. Resonant gap opening 129 5.1.1. “Spider” decorations 133 CONTENTS ix

5.2. Zeros of eigenfunctions and nodal domains 134 5.2.1. Some basic results 137 5.2.2. Bounds on the nodal count 138 5.2.3. Nodal count for special types of graphs 142 5.2.4. Nodal deﬁciency and Morse indices 143 5.3. Spectral determinants of quantum graphs 148 5.4. Scattering on quantum graphs 150 5.5. Further remarks and references 153

Chapter 6. Quantum Chaos on Graphs 157 6.1. Classical “motion” on graphs 158 6.2. Spectral statistics and random matrix theory 159 6.2.1. Form factor of a unitary matrix 161 6.2.2. Random matrices 162 6.3. Spectral statistics of graphs 164 6.4. Periodic orbit expansions 167 6.4.1. On time-reversal invariance 170 6.4.2. Diagonal approximation 170 6.4.3. The simplest example of an oﬀ-diagonal term 173 6.5. Further remarks and references 179

Chapter 7. Some Applications and Generalizations 181 7.1. Inverse problems 181 7.1.1. Can one hear the shape of a quantum graph? 183 7.1.2. Quantum graph isospectrality 184 7.1.3. Can one count the shape of a graph? 185 7.1.4. Inverse scattering 185 7.1.5. Discrete “electrical impedance” problem 185 7.2. Other types of equations on metric graphs 186 7.2.1. Heat equation 186 7.2.2. Wave equation 186 7.2.3. Control theory 186 7.2.4. Reaction-diffusion equations 187 7.2.5. Dirac and Rashba operators 187 7.2.6. Pseudo-differential Hamiltonians 187 7.2.7. Non-linear Schrödinger equation (NLS) 188 7.3. Analysis on fractals 188 7.4. Equations on multistructures 188 7.5. Graph models of thin structures 188 7.5.1. Neumann tubes 190 7.5.2. Dirichlet tubes 192 7.5.3. “Leaky” structures 194 xCONTENTS

7.6. Quantum graph modeling of various physical phenomena 200 7.6.1. Simulation of quantum graphs by microwave networks 200 7.6.2. Realizability questions 200 7.6.3. Spectra of graphene and carbon nanotubes 201 7.6.4. Vacuum energy and Casimir effect 205 7.6.5. Anderson localization 207 7.6.6. Bose-Einstein condensates 207 7.6.7. Quantum Hall effect 207 7.6.8. Flat band phenomena and slowing down light 208 Appendix A. Some Notions of Graph Theory 209 A.1. Graph, edge, vertex, degree 209 A.2. Some special graphs 210 A.3. Graphs and digraphs 210 A.4. Paths, closed paths, Betti number 211 A.5. Periodic graph 211 A.6. Cayley graphs and Schreier graphs 212 Appendix B. Linear Operators and Operator-Functions 213 B.1. Some notation concerning linear operators 213 B.2. Fredholm and semi-Fredholm operators. Fredholm index 213 B.3. Analytic Fredholm operator functions 215 B.3.1. Some notions from the several complex variables theory 215 B.3.2. Analytic Fredholm operator functions 216 Appendix C. Structure of Spectra 219 C.1. Classification of the points of the spectrum 220 C.2. Spectral theorem and spectrum classification 220 Appendix D. Symplectic Geometry and Extension Theory 223 Bibliography 227 Index 267 Preface

In this book, the name “quantum graph” refers to a graph considered as a one-dimensional simplicial complex and equipped with a differential operator (“Hamiltonian”). Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s in various areas of chemistry, physics, and mathematics. However, as a coherent and actively pursued topic, the area of quantum graphs has experienced an explosive growth only in the last couple of decades. There are manifold reasons for this surge. Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., “meso-” or “nano-scale”) system that looks like a thin neighborhood of a graph. One can mention in particular the free-electron theory of conjugated molecules, quantum wires, photonic crystals, carbon nano-structures, thin waveguides, some problems of dynamical systems, system theory and number theory, and many other applications that have led inde- pendently to quantum graph models. Quantum graphs also play a role of simplified, although still non-trivial, models for studying difficult issues, for instance, Anderson localization and quantum chaos. There are fruitful relations of quantum graphs with the older spectral theory of “standard” (combinatorial) graphs [191, 195, 213–215, 415] and with what is sometimes called discrete geometric analysis [682]. Quantum graphs present new non-trivial mathematical challenges, which makes them dear to a mathematician’s heart. As the reader will see, work on quantum graph theory and applications has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. In the new millennium, these relations between the various topics leading to quantum graphs were noticed, which has triggered a series of interdisciplinary meetings and intensive communication and coop- eration among researchers coming from different areas of science and engineering. Surveys and collections of papers on quantum graphs and related issues have started to appear (e.g., [98,121,122,126,161,353,

xi xii PREFACE

421,438,477–480,536,614,615,622]). These surveys, however, usu- ally focus on special features of quantum graph theory and there is still no comprehensive introduction to the topic, which is why the authors decided to write this text. The book is intended to serve a dual pur- pose: to provide an introduction to and survey of the current state of quantum graph theory, as well as to serve as a reference text, where the main notions and techniques are collected. The authors are indebted to many colleagues, from whom they have learned over the years a great deal about quantum graph theory and related issues. This includes, in particular, M. Aizenman, R. Band, J. Bolte, R. Carlson, Y. Colin de Verdiere, P. Exner, A. Figotin, M. Freidlin, L. Friedlander, S. Fulling, S. Gnutzmann, R. Grigorchuk, J.M. Harrison, J. P. Keating, E. Korotyaev, V. Kostrykin, M. Kotani, T. Kottos, L. Kunyansky, S. Molchanov, V. Nekrashevich, S. Novikov, K. Pankrashkin, B. Pavlov, O. Post, H. Schantz, R. Schrader, M. Shu- bin, U. Smilansky, A. Sobolev, M. Solomyak, T. Sunada, A. Teplyaev, B. Vainberg, I. Veselić, and B. Winn. The second author has been sig- nificantly influenced by the late M. Birman, R. Brooks, and V. Geyler. We cordially thank S. Fulling, W. Justice and our graduate students N. Do, W. Liu and T. Weyand for their critical reading of the manu- script and numerous corrections. We are grateful to the reviewers, especially those who had to endure the cruel and unusual punishment of refereeing several versions, for many important suggestions. Our thanks go to the people from the AMS office for invaluable help. We are especially grateful to S. Gelfand, without whose insistence this book would have never been finished, and C. Thivierge for constant support during the production. Some parts of this work were supported by grants from the National Science Foundation. The authors express their gratitude to the NSF for the support. Finally, we are grateful to our families for their limitless patience.

Gregory Berkolaiko and Peter Kuchment College Station, TX July 2012 Introduction

In this book, our goal is to introduce the main notions, structures, and techniques used in quantum graph studies, as well as to provide a brief survey of more special topics and applications. This task has shaped the book as follows: we present in detail the basic constructions and frequently used technical results in Chapters 1 and 2, devoted to quantum graph operators, and Chapters 3 – 5, which address various issues of the spectral theory of quantum graphs. The remaining two chapters are of review nature and thus less detailed; in most cases the reader will be directed to the cited literature for precise formulations and proofs. Using graphs as models for quantum chaos is considered in Chapter 6. Chapter 7 provides a brief survey of various generalizations and applications. The reader will notice that the area is developing very fast; had we tried to be more specific in this chapter, it would be outdated by the time of publications anyway. Our intent was to make the book accessible to graduate and advanced undergraduate students in mathematics, physics, and engineering. Since a variety of techniques are used, for the benefit of the reader we introduce the main notions and relevant results in graph theory, functional analysis, and operator theory in a series of Appendices. In order to make reading smoother, we normally do not include references in the main text of the chapters, collecting them, as well as additional comments, in the specially devoted last section of each chapter. We also have not tried to make the considerations too general. For instance, we mostly treat the second derivative operators on quantum graphs, while considerations could be easily extended to the more general Schrödinger operators. When we do mention more general operators, we do not look for the most general conditions on the coefficients (potentials), settling for some reasonable conditions that make the techniques work.

xiii

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1D complex, 3, 4 automorphic, 110 B,3 E,2 band-gap structure, 116 N(λ), 67 Bethe lattices, 34, 207 V ,2 Betti number, 4, 211 W , 105, 107 Bloch variety, 116 Φ(E,F), Fredholm operators, 214 bond, 2, 210 Tn, 109 bond reversal, 3 βΓ, 211 bound state, 80 D(Γ), 111 boundary control method, 184 δ-type condition, 24 boundary form, 20 fˆ, 110 boundary vertices, 150 μ(f), 135 Brillouin zone, 109 ν(f), 135 π-electrons, 192 carbon nanotube, 201 ρ(A), 219 Casimir effect, 205 σ-electrons, 192 Casimir-Polder force, 205 σd(A), 220 Cayley graph, 28, 35, 212 σac(A), 221 character, 107 σpp(A), 221 chiral, 204 σr(A), 220 co-boundary operator, 7 σsc(A), 221 co-spectral graphs, 183 σ(A), 219 co-spectrum, 216 σess(A), 220 combinatorial graph, 3 H k(Γ), 11 condition δ-type, 24 du,2 i o continuity, 14 dv,dv,3 1st Betti number, 4, 211 current conservation, 14 decoupling, 26 adjacency matrix, 2, 209 Dirichlet, 25 adjacent vertices, 2 Kirchhoff, 14 algebraic counting function, 93 Neumann, 14 analysis on fractals, 188 Neumann–Kirchhoff, 14 analytic Fredholm theorem, 67 quasiperiodic, 60, 114 analytic set, 216 standard, 14 aperiodic graph, 172 strongly coupling, 172 arcs, 175 continuous functional calculus, 220 267 268 INDEX critical point, 147 forest, 211 non-degenerate, 147 form CROWs, 208 closed, 219 current conservation condition, 14 Hermitian, 219 cut-points, 60 Hermitian symplectic, 223 cycle, 211 non-degenerate, 223 cyclomatic number, 4, 211 quadratic, 219 quadratic, of a graph, 22 decoupling condition, 26 sesqui-linear, 219 degeneracy class, 168 form factor, 161 degree matrix, 2, 209 Fredholm operator, 52, 213 degree of a vertex fullerene buckyball, 201 outgoing, 210 function of a discrete graph, 5 degree of a vertex, 2, 209 fundamental domain, 105, 107 incoming, 210 density of states, 100, 160, 221 Gaussian Orthogonal Ensemble determinant, 149 (GOE), 162 diagrams, 179 Gaussian Unitary Ensemble, 163 digraph, 2, 210 graph, 2, 209 Dirac operators, 187 bipartite, 210 Dirac point, 203 compact, 9 directed adjacency matrix, 210 complete, 210 directed graph, 2, 210 finite, 9 Dirichlet-to-Neumann map, 85, 87, infinite, 9 198, 201 periodic, 105, 211 Discrete Fourier Transform (DFT) regular, 210 matrix, 50 star, 210 discrete graph, 3 tree, 211 dispersion curve, 116 graphene, 201 dispersion relation, 68, 116 graphyne, 201 Grassmannian, 224 eigenfunction, 220 eigenvalue, 220 Hadamard’s variational formulas, 68 electric potential, 12 Hamiltonian, 12, 29 electromagnetic circuits, 188 Hermitian symplectic space, 223 ellipticity, 53 Hill discriminant, 201 encounter, 175 equilateral graph, 8, 90 improper partition vertex, 143 equipartition, 144 impurity, 124 equitransmitting matrices, 50 impurity spectrum, 124 extension theory, 223 incidence matrix, 6 external edges, 150 incoming amplitude, 150 incoming bond, 3, 210 fattened graph, 189 incoming degree, 3 finite ball condition, 9 index of a Fredholm operator, 52, Floquet multiplier, 109 214 Floquet theory, 105 Internet (or network) tomography, Floquet transform, 109 182 flux, 61 inverse problem, 181 INDEX 269 invertible regularizer, 214 nodal, 143 invisibility cloaking, 195 proper, 143 isospectral graphs, 183 partition vertex, 143 isotropic subspace, 224 path, 99, 211 closed, 99, 211 Lagrangian subspace, 224 primitive, 99 Laplacian repetition, 99 combinatorial, 5 perfect scar, 85 discrete, 5 periodic orbit, 99 harmonic, 5 photonic crystal, 195 normalized, 6 positive (negative) domain, 135 weighted, 7 potential, 203 Laurent polynomials, 112 principal analytic, 68 lead, 8, 150 principal analytic set, 216 leaky wire, 195 proper partition vertex, 143 left regularizer, 215 left semi-Fredholm, 215 quadratic form, 5 level repulsion, 159 quantum graph, 3, 13 Liouville type theorem, 127 quantum waveguide, 193 local finiteness, 2, 209 quantum wire, 193 local vertex conditions, 16 quasi-momentum, 108 Lyapunov function, 201 quasiperiodic conditions, 61 magnetic potential, 12 Rashba Hamiltonian, 187 Markov operator, 6 regularized determinant, 101 metric graph, 3, 7 regularizer, 214 metric length, 100 repetition number, 99 Morse index, 147 repetition of a closed path, 99 multistructure, 188 resolvent, 219 resolvent set, 219 naphthalene, 192 right regularizer, 215 necklace structure, 204 right semi-Fredholm, 215 nodal count, 135 nodal deficiency, 145 scale invariance, 22 nodal domain, 134 scar, 85 number of zeros, 135 scattering matrix, 38, 150 Oka’s principle, 215 Schatten-von Neumann operator operator pencils, 49 ideal, 101 origin vertex, 2, 210 Schreier graph, 28, 35, 212 outgoing, 3 SCISSORs, 208 outgoing amplitudes, 150 semi-Fredholm operators, 215 outgoing bond, 3, 210 simple cycle, 211 outgoing degree, 3 simplicial complex, xi, 8 outgoing directions, 14 simplicity of the spectrum generic, 71 partition, 143 of a tree, 74 bipartite, 144 skew orthogonal, 224 equipartition, 144 Sobolev space H1(Γ), 10 improper, 143 space L2(Γ), 10 270 INDEX spectral band, 115 spectral counting function, 67 spectral determinant, 148 spectral gap, 115 spectral measure, 221 spectral theorem, 220 spectrum, 219 absolutely continuous, 221 discrete, 220 essential, 220 point, 220 pure point, 221 residual, 220 singular continuous, 221 spectrum of the operator function, 216 stability amplitude, 100 Stein domain, 55 Stein manifold, 215, 216 strongly coupling vertex conditions, 182 symplectic geometry, 223 system theory, 201 terminal vertex, 2, 210 threshold, 194 time-reversal invariance, 170 topological length, 99 trace formula, 92 tree, 4, 211 two-point correlation function, 160 unique continuation property, 84 unitary character, 107 vertex conditions, 13 vertex Neumann conditions, 27 weak unique continuation property, 84 Selected Published Titles in This Series

186 Gregory Berkolaiko and Peter Kuchment, Introduction to Quantum Graphs, 2013 185 Patrick Iglesias-Zemmour, Diffeology, 2012 184 Frederick W. Gehring and Kari Hag, The Ubiquitous Quasidisk, 2012 183 Gershon Kresin and Vladimir Maz’ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, 2012 182 Neil A. Watson, Introduction to Heat Potential Theory, 2012 181 Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay Representations, 2012 180 Martin W. Liebeck and Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, 2012 179 Stephen D. Smith, Subgroup Complexes, 2011 178 Helmut Brass and Knut Petras, Quadrature Theory, 2011 177 Alexei Myasnikov, Vladimir Shpilrain, and Alexander Ushakov, Non-commutative Cryptography and Complexity of Group-theoretic Problems, 2011 176 Peter E. Kloeden and Martin Rasmussen, Nonautonomous Dynamical Systems, 2011 175 Warwick de Launey and Dane Flannery, Algebraic Design Theory, 2011 174 Lawrence S. Levy and J. Chris Robson, Hereditary Noetherian Prime Rings and Idealizers, 2011 173 Sariel Har-Peled, Geometric Approximation Algorithms, 2011 172 Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon, The Classification of Finite Simple Groups, 2011 171 Leonid Pastur and Mariya Shcherbina, Eigenvalue Distribution of Large Random Matrices, 2011 170 Kevin Costello, Renormalization and Effective Field Theory, 2011 169 Robert R. Bruner and J. P. C. Greenlees, Connective Real K-Theory of Finite Groups, 2010 168 Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, Rings and Modules, 2010 167 Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster Algebras and Poisson Geometry, 2010 166 Kyung Bai Lee and Frank Raymond, Seifert Fiberings, 2010 165 Fuensanta Andreu-Vaillo, JoséM.Mazón, Julio D. Rossi, and J. Julián Toledo-Melero, Nonlocal Diffusion Problems, 2010 164 Vladimir I. Bogachev, Differentiable Measures and the Malliavin Calculus, 2010 163 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic Aspects, 2010 162 Vladimir Mazya and Jürgen Rossmann, Elliptic Equations in Polyhedral Domains, 2010 161 KanishkaPerera,RaviP.Agarwal,andDonalO’Regan, Morse Theoretic Aspects of p-Laplacian Type Operators, 2010 160 Alexander S. Kechris, Global Aspects of Ergodic Group Actions, 2010 159 Matthew Baker and Robert Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, 2010 158 D. R. Yafaev, Mathematical Scattering Theory, 2010 157 Xia Chen, Random Walk Intersections, 2010 156 Jaime Angulo Pava, Nonlinear Dispersive Equations, 2009 155 Yiannis N. Moschovakis, Descriptive Set Theory, Second Edition, 2009

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/.

A “quantum graph” is a graph considered as a one-dimensional complex and equipped with a differential operator (“Hamiltonian”). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., “meso-” Subbarao Vani Photo courtesy of or “nano-scale”) system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician’s heart. Work on quantum graphs has brought together tools and intuition coming from graph theory, combinatorics, mathematical physics, PDEs, and spectral theory. This book provides a comprehensive introduction to the topic, collecting the main notions and techniques. It also contains a survey of the current state of the quantum graph research and applications.

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