“量子图”被认为是一维复合体，并配备了微分算子（“Hamilton算子”）。当人们考虑各种波通过类似于图的薄邻域的准一维（例如“中等尺度”或“纳米尺度”）系统传播时，量子图在数学、物理、化学和工程中自然而然地作为简化模型出现。至少从 20世纪30年代开始，有关量子图的研究已经出现，从那时起，量子图技术已经成功地应用于数学物理、一般数学及其应用的各个领域。例如，动力系统理论、控制理论、量子混沌、Anderson局域化、微电子学、光子晶体、物理化学、纳米科学、超导理论等。 量子图提出了许多非平凡的数学挑战，这使得它们成为数学家钟爱的对象。量子图的研究汇集了来自图论、组合学、数学物理、偏微分方程和谱理论等领域的工具和直觉。 本书全面介绍了这个主题，收集了主要的概念和技术。它还包括对当前量子图研究和应用状况的概述。

前辅文
Preface
Introduction
Chapter 1. Operators on Graphs. Quantum graphs
  1.1. Main graph notions and notation
  1.2. Difference operators. Discrete Laplace operators
  1.3. Metric graphs
  1.4. Differential operators on metric graphs. Quantum graphs
   1.4.1. Vertex conditions. Finite graphs.
   1.4.2. Scale invariance
   1.4.3. Quadratic form
   1.4.4. Examples of vertex conditions
   1.4.5. Infinite graphs
   1.4.6. Non-local vertex conditions
  1.5. Further remarks and references
Chapter 2. Quantum Graph Operators. Special Topics
  2.1. Quantum graphs and scattering matrices
   2.1.1. Scattering on vertices
   2.1.2. Bond scattering matrix and the secular equation
  2.2. First order operators and scattering matrices
  2.3. Factorization of quantum graph Hamiltonians
  2.4. Index of quantum graph operators
  2.5. Dependence on vertex conditions
   2.5.1. Variations in the edge lengths
  2.6. Magnetic Schrödinger operator
  2.7. Further remarks and references
Chapter 3. Spectra of Quantum Graphs
  3.1. Basic spectral properties of compact quantum graphs
   3.1.1. Discreteness of the spectrum
   3.1.2. Dependence on the vertex conditions
   3.1.3. Eigenfunction dependence
   3.1.4. An Hadamard-type formula
   3.1.5. Generic simplicity of the spectrum
   3.1.6. Eigenvalue bracketing
   3.1.7. Dependence on the coupling constant at a vertex
  3.2. The Shnol’ theorem
  3.3. Generalized eigenfunctions
  3.4. Failure of the unique continuation property. Scars
  3.5. The ubiquitous Dirichlet-to-Neumann map
   3.5.1. DtN map for a single edge
   3.5.2. DtN map for a compact graph with a “boundary”
   3.5.3. DtN map for a single vertex boundary
   3.5.4. DtN map and the secular equation
   3.5.5. DtN map and number of negative eigenvalues
  3.6. Relations between quantum and discrete graph spectra
  3.7. Trace formulas
   3.7.1. Secular equation
   3.7.2. Weyl’s law
   3.7.3. Derivation of the trace formula
   3.7.4. Expansion in terms of periodic orbits
   3.7.5. Other formulations of the trace formula
  3.8. Further remarks and references
Chapter 4. Spectra of Periodic Graphs
  4.1. Periodic graphs
  4.2. Floquet-Bloch theory
   4.2.1. Floquet transform on combinatorial periodic graphs
   4.2.2. Floquet transform of periodic difference operators
   4.2.3. Floquet transform on quantum periodic graphs
   4.2.4. Floquet transform of periodic operators
  4.3. Band-gap structure of spectrum
   4.3.1. Discrete case
   4.3.2. Quantum graph case
   4.3.3. Floquet transform in Sobolev classes
  4.4. Absence of the singular continuous spectrum
  4.5. The point spectrum
  4.6. Where do the spectral edges occur?
  4.7. Existence and location of spectral gaps
  4.8. Impurity spectra
  4.9. Further remarks and references
Chapter 5. Spectra of Quantum Graphs. Special Topics
  5.1. Resonant gap opening
   5.1.1. “Spider” decorations
  5.2. Zeros of eigenfunctions and nodal domains
   5.2.1. Some basic results
   5.2.2. Bounds on the nodal count
   5.2.3. Nodal count for special types of graphs
   5.2.4. Nodal deficiency and Morse indices
  5.3. Spectral determinants of quantum graphs
  5.4. Scattering on quantum graphs
  5.5. Further remarks and references
Chapter 6. Quantum Chaos on Graphs
  6.1. Classical “motion” on graphs
  6.2. Spectral statistics and random matrix theory
   6.2.1. Form factor of a unitary matrix
   6.2.2. Random matrices
  6.3. Spectral statistics of graphs
  6.4. Periodic orbit expansions
   6.4.1. On time-reversal invariance
   6.4.2. Diagonal approximation
   6.4.3. The simplest example of an off-diagonal term
  6.5. Further remarks and references
Chapter 7. Some Applications and Generalizations
  7.1. Inverse problems
   7.1.1. Can one hear the shape of a quantum graph?
   7.1.2. Quantum graph isospectrality
   7.1.3. Can one count the shape of a graph?
   7.1.4. Inverse scattering
   7.1.5. Discrete “electrical impedance” problem
  7.2. Other types of equations on metric graphs
   7.2.1. Heat equation
   7.2.2. Wave equation
   7.2.3. Control theory
   7.2.4. Reaction-diffusion equations
   7.2.5. Dirac and Rashba operators
   7.2.6. Pseudo-differential Hamiltonians
   7.2.7. Non-linear Schrödinger equation (NLS)
  7.3. Analysis on fractals
  7.4. Equations on multistructures
  7.5. Graph models of thin structures
   7.5.1. Neumann tubes
   7.5.2. Dirichlet tubes
   7.5.3. “Leaky” structures
  7.6. Quantum graph modeling of various physical phenomena
   7.6.1. Simulation of quantum graphs by microwave networks
   7.6.2. Realizability questions
   7.6.3. Spectra of graphene and carbon nanotubes
   7.6.4. Vacuum energy and Casimir effect
   7.6.5. Anderson localization
   7.6.6. Bose-Einstein condensates
   7.6.7. Quantum Hall effect
   7.6.8. Flat band phenomena and slowing down light
Appendix A. Some Notions of Graph Theory
  A.1. Graph, edge, vertex, degree
  A.2. Some special graphs
  A.3. Graphs and digraphs
  A.4. Paths, closed paths, Betti number
  A.5. Periodic graph
  A.6. Cayley graphs and Schreier graphs
Appendix B. Linear Operators and Operator-Functions
  B.1. Some notation concerning linear operators
  B.2. Fredholm and semi-Fredholm operators. Fredholm index
  B.3. Analytic Fredholm operator functions
  B.3.1. Some notions from the several complex variables theory
  B.3.2. Analytic Fredholm operator functions
Appendix C. Structure of Spectra
  C.1. Classification of the points of the spectrum
  C.2. Spectral theorem and spectrum classification
Appendix D. Symplectic Geometry and Extension Theory
Bibliography
Index