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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
版面字数：
450千字
开本：
特殊
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2024年
ISBN：
978-7-04-061260-8
物料号：
61260-00
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
  • 目录
    • 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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