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Introduction to Quantum Graphs [Kõva köide]

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  • Formaat: Hardback, 275 pages, kaal: 654 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Jan-2013
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821892118
  • ISBN-13: 9780821892114
Teised raamatud teemal:
Introduction to Quantum GraphsSuurem pilt
  • Formaat: Hardback, 275 pages, kaal: 654 g
  • Sari: Mathematical Surveys and Monographs
  • Ilmumisaeg: 30-Jan-2013
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821892118
  • ISBN-13: 9780821892114
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780821892114.html
Märksõnad:
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-'' 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.
Preface xi
Introduction xiii
Chapter 1 Operators on Graphs. Quantum graphs
1(36)
1.1 Main graph notions and notation
2(3)
1.2 Difference operators. Discrete Laplace operators
5(2)
1.3 Metric graphs
7(5)
1.4 Differential operators on metric graphs. Quantum graphs
12(21)
1.4.1 Vertex conditions. Finite graphs
15(7)
1.4.2 Scale invariance
22(1)
1.4.3 Quadratic form
22(2)
1.4.4 Examples of vertex conditions
24(3)
1.4.5 Infinite graphs
27(5)
1.4.6 Non-local vertex conditions
32(1)
1.5 Further remarks and references
33(4)
Chapter 2 Quantum Graph Operators. Special Topics
37(28)
2.1 Quantum graphs and scattering matrices
37(7)
2.1.1 Scattering on vertices
37(4)
2.1.2 Bond scattering matrix and the secular equation
41(3)
2.2 First order operators and scattering matrices
44(7)
2.3 Factorization of quantum graph Hamiltonians
51(1)
2.4 Index of quantum graph operators
52(2)
2.5 Dependence on vertex conditions
54(5)
2.5.1 Variations in the edge lengths
58(1)
2.6 Magnetic Schrodinger operator
59(3)
2.7 Further remarks and references
62(3)
Chapter 3 Spectra of Quantum Graphs
65(40)
3.1 Basic spectral properties of compact quantum graphs
66(13)
3.1.1 Discreteness of the spectrum
66(1)
3.1.2 Dependence on the vertex conditions
67(1)
3.1.3 Eigenfunction dependence
68(1)
3.1.4 An Hadamard-type formula
68(3)
3.1.5 Generic simplicity of the spectrum
71(1)
3.1.6 Eigenvalue bracketing
72(4)
3.1.7 Dependence on the coupling constant at a vertex
76(3)
3.2 The Shnol' theorem
79(3)
3.3 Generalized eigenfunctions
82(2)
3.4 Failure of the unique continuation property. Scars
84(1)
3.5 The ubiquitous Dirichlet-to-Neumann map
85(5)
3.5.1 DtN map for a single edge
85(2)
3.5.2 DtN map for a compact graph with a "boundary"
87(2)
3.5.3 DtN map for a single vertex boundary
89(1)
3.5.4 DtN map and the secular equation
89(1)
3.5.5 DtN map and number of negative eigenvalues
90(1)
3.6 Relations between quantum and discrete graph spectra
90(2)
3.7 Trace formulas
92(9)
3.7.1 Secular equation
93(2)
3.7.2 Weyl's law
95(1)
3.7.3 Derivation of the trace formula
96(3)
3.7.4 Expansion in terms of periodic orbits
99(1)
3.7.5 Other formulations of the trace formula
100(1)
3.8 Further remarks and references
101(4)
Chapter 4 Spectra of Periodic Graphs
105(24)
4.1 Periodic graphs
105(2)
4.2 Floquet-Bloch theory
107(8)
4.2.1 Floquet transform on combinatorial periodic graphs
109(3)
4.2.2 Floquet transform of periodic difference operators
112(1)
4.2.3 Floquet transform on quantum periodic graphs
113(1)
4.2.4 Floquet transform of periodic operators
114(1)
4.3 Band-gap structure of spectrum
115(2)
4.3.1 Discrete case
115(1)
4.3.2 Quantum graph case
116(1)
4.3.3 Floquet transform in Sobolev classes
117(1)
4.4 Absence of the singular continuous spectrum
117(1)
4.5 The point spectrum
118(3)
4.6 Where do the spectral edges occur?
121(2)
4.7 Existence and location of spectral gaps
123(1)
4.8 Impurity spectra
124(1)
4.9 Further remarks and references
124(5)
Chapter 5 Spectra of Quantum Graphs. Special Topics
129(28)
5.1 Resonant gap opening
129(5)
5.1.1 "Spider" decorations
133(1)
5.2 Zeros of eigenfunctions and nodal domains
134(14)
5.2.1 Some basic results
137(1)
5.2.2 Bounds on the nodal count
138(4)
5.2.3 Nodal count for special types of graphs
142(1)
5.2.4 Nodal deficiency and Morse indices
143(5)
5.3 Spectral determinants of quantum graphs
148(2)
5.4 Scattering on quantum graphs
150(3)
5.5 Further remarks and references
153(4)
Chapter 6 Quantum Chaos on Graphs
157(24)
6.1 Classical "motion" on graphs
158(1)
6.2 Spectral statistics and random matrix theory
159(5)
6.2.1 Form factor of a unitary matrix
161(1)
6.2.2 Random matrices
162(2)
6.3 Spectral statistics of graphs
164(3)
6.4 Periodic orbit expansions
167(12)
6.4.1 On time-reversal invariance
170(1)
6.4.2 Diagonal approximation
170(3)
6.4.3 The simplest example of an off-diagonal term
173(6)
6.5 Further remarks and references
179(2)
Chapter 7 Some Applications and Generalizations
181(28)
7.1 Inverse problems
181(5)
7.1.1 Can one hear the shape of a quantum graph?
183(1)
7.1.2 Quantum graph isospectrality
184(1)
7.1.3 Can one count the shape of a graph?
185(1)
7.1.4 Inverse scattering
185(1)
7.1.5 Discrete "electrical impedance" problem
185(1)
7.2 Other types of equations on metric graphs
186(2)
7.2.1 Heat equation
186(1)
7.2.2 Wave equation
186(1)
7.2.3 Control theory
186(1)
7.2.4 Reaction-diffusion equations
187(1)
7.2.5 Dirac and Rashba operators
187(1)
7.2.6 Pseudo-differential Hamiltonians
187(1)
7.2.7 Non-linear Schrodinger equation (NLS)
188(1)
7.3 Analysis on fractals
188(1)
7.4 Equations on multistructures
188(1)
7.5 Graph models of thin structures
188(12)
7.5.1 Neumann tubes
190(2)
7.5.2 Dirichlet tubes
192(2)
7.5.3 "Leaky" structures
194(6)
7.6 Quantum graph modeling of various physical phenomena
200(9)
7.6.1 Simulation of quantum graphs by microwave networks
200(1)
7.6.2 Realizability questions
200(1)
7.6.3 Spectra of graphene and carbon nanotubes
201(4)
7.6.4 Vacuum energy and Casimir effect
205(2)
7.6.5 Anderson localization
207(1)
7.6.6 Bose-Einstein condensates
207(1)
7.6.7 Quantum Hall effect
207(1)
7.6.8 Flat band phenomena and slowing down light
208(1)
Appendix A Some Notions of Graph Theory
209(4)
A.1 Graph, edge, vertex, degree
209(1)
A.2 Some special graphs
210(1)
A.3 Graphs and digraphs
210(1)
A.4 Paths, closed paths, Betti number
211(1)
A.5 Periodic graph
211(1)
A.6 Cayley graphs and Schreier graphs
212(1)
Appendix B Linear Operators and Operator-Functions
213(6)
B.2 Some notation concerning linear operators
213(1)
B.2 Fredholm and semi-Fredholm operators. Fredholm index
213(2)
B.3 Analytic Fredholm operator functions
215(1)
B.3.1 Some notions from the several complex variables theory
215(1)
B.3.2 Analytic Fredholm operator functions
216(3)
Appendix C Structure of Spectra
219(4)
C.1 Classification of the points of the spectrum
220(1)
C.2 Spectral theorem and spectrum classification
220(3)
Appendix D Symplectic Geometry and Extension Theory
223(4)
Bibliography 227(40)
Index 267