Spectral Geometry of Graphs / / by Pavel Kurasov.

• Superior document: Operator Theory: Advances and Applications, 293.
• Author: Kurasov, Pavel.
• Place / Publishing House: Berlin, Heidelberg : : Springer Berlin Heidelberg :, Imprint: Birkhäuser,, 2024.
• Year of Publication: 2024
• Edition: First edition 2024.
• Language: English
• Series: Operator Theory: Advances and Applications, 293
• Physical Description: 1 online resource (0 pages)

This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.

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.