Spectral Geometry of Graphs.

• Superior document: Operator Theory: Advances and Applications Series ; v.293
• Author: 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
• Physical Description: 1 online resource (644 pages)

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.