Spectral Geometry of Graphs.

Spectral Geometry of Graphs.

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Bibliographic Details
Superior document:Operator Theory: Advances and Applications Series ; v.293
:
Kurasov, Pavel.
Place / Publishing House:Berlin, Heidelberg : : Springer Basel AG,, 2023.
©2024.
Year of Publication:2023
Edition:1st ed.
Language:English
Series:Operator Theory: Advances and Applications Series
Online Access:
Physical Description:1 online resource (644 pages)
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Table of Contents:
  • Intro
  • Notations
  • Conventions
  • Contents
  • 1 Very Personal Introduction
  • 2 How to Define Differential Operators on Metric Graphs
  • 2.1 Schrödinger Operators on Metric Graphs
  • 2.1.1 Metric Graphs
  • 2.1.2 Differential Operators
  • 2.1.3 Standard Vertex Conditions
  • 2.1.4 Definition of the Operator
  • 2.2 Elementary Examples
  • 3 Vertex Conditions
  • 3.1 Preliminary Discussion
  • 3.2 Vertex Conditions for the Star Graph
  • 3.3 Vertex Conditions Via the Vertex Scattering Matrix
  • 3.3.1 The Vertex Scattering Matrix
  • 3.3.2 Scattering Matrix as a Parameterin the Vertex Conditions
  • 3.3.3 On Properly Connecting Vertex Conditions
  • 3.4 Parametrisation Via Hermitian Matrices
  • 3.5 Scaling-Invariant and Standard Conditions
  • 3.5.1 Energy Dependence of the Vertex S-matrix
  • 3.5.2 Scaling-Invariant, or Non-Robin Vertex Conditions
  • 3.5.3 Standard Vertex Conditions
  • 3.6 Signing Conditions for Degree Two Vertices
  • 3.7 Generalised Delta Couplings
  • 3.8 Vertex Conditions for Arbitrary Graphs and Definition of the Magnetic Schrödinger Operator
  • 3.8.1 Scattering Matrix Parametrisationof Vertex Conditions
  • 3.8.2 Quadratic Form Parametrisation of Vertex Conditions
  • Appendix 1: Important Classes of Vertex Conditions
  • δ and δ'-Couplings
  • Circulant Conditions
  • `Real' Conditions
  • Indistinguishable Edges
  • Equi-transmitting Vertices
  • Appendix 2: Parametrisation of Vertex Conditions: Historical Remarks
  • Parametrisation Via Linear Relations
  • Parametrisation Using Hermitian Operators
  • Unitary Matrix Parametrisation
  • 4 Elementary Spectral Properties of Quantum Graphs
  • 4.1 Quantum Graphs as Self-adjoint Operators
  • 4.2 The Dirichlet Operator and the Weyl's Law
  • 4.3 Spectra of Quantum Graphs
  • 4.4 Laplacian Ground State
  • 4.5 Bonus Section: Positivity of the Ground Statefor Quantum Graphs.
  • 4.5.1 The Case of Standard Vertex Conditions
  • 4.5.2 A Counterexample
  • 4.5.3 Invariance of the Quadratic Form
  • 4.5.4 Positivity of the Ground State for Generalised Delta-Couplings
  • 4.6 First Spectral Estimates
  • 5 The Characteristic Equation
  • 5.1 Characteristic Equation I: Edge Transfer Matrices
  • 5.1.1 Transfer Matrix for a Single Interval
  • One-Dimensional Schrödinger Equation
  • Magnetic Schrödinger Equation
  • 5.1.2 The Characteristic Equation
  • 5.1.3 The Characteristic Equation, Second Look
  • 5.2 Characteristic Equation II: Scattering Approach
  • 5.2.1 On the Scattering Matrix Associated with a Compact Interval
  • 5.2.2 Positive Spectrum and Scattering Matrices for Finite Compact Graphs
  • 5.3 Characteristic Equation III: M-Function Approach
  • 5.3.1 M-Function for a Single Interval
  • 5.3.2 The Edge M-Function
  • 5.3.3 Characteristic Equation via the M-Function: General Vertex Conditions
  • 5.3.4 Reduction of the M-Function for Standard Vertex Conditions
  • 6 Standard Laplacians and Secular Polynomials
  • 6.1 Secular Polynomials
  • 6.2 Secular Polynomials for Small Graphs
  • 6.3 Zero Sets for Small Graphs
  • Appendix 1: Singular Sets on Secular Manifolds, Proof of Lemma 6.3
  • 7 Reducibility of Secular Polynomials
  • 7.1 Contraction of Graphs
  • 7.2 Extensions of Graphs
  • 7.3 Secular Polynomials for the Watermelon Graphand Its Closest Relatives
  • 7.4 Secular Polynomials for Flower Graphs and Their Extensions
  • 7.5 Reducibility of Secular Polynomials for General Graphs
  • 8 The Trace Formula
  • 8.1 The Characteristic Equation: Multiplicityof Positive Eigenvalues
  • 8.2 Algebraic and Spectral Multiplicities of the Eigenvalue Zero
  • 8.3 The Trace Formula for Standard Laplacians
  • 8.4 Trace Formula for Laplacians with Scaling-InvariantVertex Conditions
  • 9 Trace Formula and Inverse Problems.
  • 9.1 Euler Characteristic for Standard Laplacians
  • 9.2 Euler Characteristic for Graphs with Dirichlet Vertices
  • 9.3 Spectral Asymptotics and Schrödinger Operators
  • 9.3.1 Euler Characteristic and Spectral Asymptotics
  • 9.3.2 Schrödinger Operators and Euler Characteristic of Graphs
  • 9.3.3 General Vertex Conditions: A Counterexample
  • 9.4 Reconstruction of Graphs with RationallyIndependent Lengths
  • 10 Arithmetic Structure of the Spectrumand Crystalline Measures
  • 10.1 Arithmetic Structure of the Spectrum
  • 10.2 Crystalline Measures
  • 10.3 The Lasso Graph and Crystalline Measures
  • 10.4 Graph's Spectrum as a Delone Set
  • 11 Quadratic Forms and Spectral Estimates
  • 11.1 Quadratic Forms (Integrable Potentials)
  • 11.1.1 Explicit Expression
  • 11.1.2 An Elementary Sobolev Estimate
  • 11.1.3 The Perturbation Term Is Form-Bounded
  • 11.1.4 The Reference Laplacian
  • 11.1.5 Closure of the Perturbed Quadratic Form
  • 11.2 Spectral Estimates (Standard Vertex Conditions)
  • 11.3 Spectral Estimates for General Vertex Conditions
  • 12 Spectral Gap and Dirichlet Ground State
  • 12.1 Fundamental Estimates
  • 12.1.1 Eulerian Path Technique
  • 12.1.2 Symmetrisation Technique
  • 12.2 Balanced and Doubly Connected Graphs
  • 12.3 Graphs with Dirichlet Vertices
  • 12.4 Cheeger's Approach
  • 12.5 Topological Perturbations in the Case of Standard Conditions
  • 12.5.1 Gluing Vertices Together
  • 12.5.2 Adding an Edge
  • 12.6 Bonus Section: Further Topological Perturbations
  • 12.6.1 Cutting Edges
  • 12.6.2 Deleting Edges
  • 13 Higher Eigenvalues and Topological Perturbations
  • 13.1 Fundamental Estimates for Higher Eigenvalues
  • 13.1.1 Lower Estimates
  • 13.1.2 Upper Bounds
  • 13.1.3 Graphs Realising Extremal Eigenvalues
  • 13.2 Gluing and Cutting Vertices with Standard Conditions
  • 13.3 Gluing Vertices with Scaling-Invariant Conditions.
  • 13.3.1 Scaling-Invariant Conditions Revisited
  • 13.3.2 Gluing Vertices
  • Gluing Vertices with One-Dimensional Vertex Conditions
  • Gluing Vertices with Hyperplanar Vertex Conditions
  • 13.3.3 Spectral Gap and Gluing Vertices with Scaling-Invariant Conditions
  • 13.4 Gluing Vertices with General Vertex Conditions
  • 14 Ambartsumian Type Theorems
  • 14.1 Two Parameters Fixed, One Parameter Varies
  • 14.1.1 Zero Potential Is Exceptional: Classical Ambartsumian Theorem
  • 14.1.2 Interval-Graph Is Exceptional: Geometric Version of Ambartsumian Theorem for Standard Laplacians
  • 14.1.3 Standard Vertex Conditions Are Not Exceptional
  • 14.2 One Parameter Is Fixed, Two Parameters Vary
  • 14.2.1 Standard Vertex Conditions Are Exceptional: Schrödinger Operators on Arbitrary Graphs
  • 14.2.2 Zero Potential: Laplacians on Graphs that Are Isospectral to the Interval
  • 14.2.3 Single Interval: Schrödinger Operators Isospectral to the Standard Laplacian
  • Crum's Procedure
  • Inverting Crum's Procedure
  • 15 Further Theorems Inspired by Ambartsumian
  • 15.1 Ambartsumian-Type Theorem by Davies
  • 15.1.1 On a Sufficient Condition for the Potential to Be Zero
  • 15.1.2 Laplacian Heat Kernel
  • Heat Kernel for the Dirichlet Laplacian on an Interval
  • Heat Kernel for the Standard Laplacian on the Graph
  • 15.1.3 On Schrödinger Semigroups
  • 15.1.4 A Theorem by Davies
  • 15.2 On Asymptotically Isospectral Quantum Graphs
  • 15.2.1 On the Zeroes of Generalised TrigonometricPolynomials
  • 15.2.2 Asymptotically Isospectral Quantum Graphs
  • 15.2.3 When a Schrödinger Operator Is Isospectral to a Laplacian
  • 16 Magnetic Fluxes
  • 16.1 Unitary Transformations via Multiplications and Magnetic Schrödinger Operators
  • 16.2 Vertex Phases and Transition Probabilities
  • 16.3 Topological Damping of Aharonov-Bohm Effect
  • 16.3.1 Getting Started.
  • 16.3.2 Explicit Calculation of the Spectrum
  • 16.3.3 Topological Reasons for Damping
  • 17 M-Functions: Definitions and Examples
  • 17.1 The Graph M-Function
  • 17.1.1 Motivation and Historical Hints
  • 17.1.2 The Formal Definition
  • 17.1.3 Examples
  • 17.2 Explicit Formulas Using Eigenfunctions
  • 17.3 Hierarchy of M-Functions for Standard Vertex Conditions
  • 18 M-Functions: Properties and First Applications
  • 18.1 M-Function as a Matrix-Valued Herglotz-Nevanlinna Function
  • 18.2 Gluing Procedure and the Spectral Gap
  • 18.2.1 Examples
  • 18.3 Gluing Graphs and M-Functions
  • 18.3.1 The M-Function for General Vertex Conditions at the Contact Set
  • 18.3.2 Gluing Graphs with General Vertex Conditions
  • Appendix 1: Scattering from Compact Graphs
  • 19 Boundary Control: BC-Method
  • 19.1 Inverse Problems: First Look
  • 19.2 How to Use BC-Method for Graphs
  • 19.3 The Response Operator and the M-Function
  • 19.4 Inverse Problem for the One-DimensionalSchrödinger Equation
  • 19.5 BC-Method for the Standard Laplacian on the Star Graph
  • 19.6 BC-Method for the Star Graph with General Vertex Conditions
  • 20 Inverse Problems for Trees
  • 20.1 Obvious Ambiguities and Limitations
  • 20.2 Subproblem I: Reconstruction of the Metric Tree
  • 20.2.1 Global Reconstruction of the Metric Tree
  • 20.2.2 Local Reconstruction of the Metric Tree
  • 20.3 Subproblem II: Reconstruction of the Potential
  • 20.4 Subproblem III: Reconstruction of the Vertex Conditions
  • 20.4.1 Trimming a Bunch
  • 20.4.2 Recovering the Vertex Conditions for an Equilateral Bunch
  • 20.5 Cleaning and Pruning Using the M-functions
  • 20.5.1 Cleaning the Edges
  • 20.5.2 Pruning Branches and Bunches
  • 20.6 Complete Solution of the Inverse Problem for Trees
  • Appendix 1: Calculation of the M-function for the Cross Graph
  • Appendix 2: Calderón Problem.
  • 21 Boundary Control for Graphs with Cycles: Dismantling Graphs.

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