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量子图导论（影印版）

样章
作者：
Gregory Berkolaiko，Peter Kuchment
定价：
135.00元
ISBN：
978-7-04-061260-8
版面字数：
450.000千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2024-02-08
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
数学应用
暂无
  • 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

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