From Complex Analysis to Operator Theory: A Panorama : In Memory of Sergey Naboko

Contents
Sergey Naboko: A Life in Mathematics
Sergey Naboko: A Life in Mathematics
	Academic Career
	Personality
	University Professor
	Mathematical Research
	Working with Sergey
	Sergey\'s Students
	The Swedish Connection
	Malcolm Brown (1946–2022)
	References
Curriculum Vitae
Curriculum Vitae
	Editorial Board for International Journals
	Referee for Journals (Selected)
	Organised Conferences
List of Publications
List of Publications
	Research Papers
	Research Papers on the Way
	Editorial Work
Photographs (Private and Academic Life)
Photographs (Private and Academic Life)
Part I On Some of Sergey\'s Works
	Working with Sergey Naboko on Boundary Triples
		1 Introduction
		2 Background
		3 Elliptic PDEs and Dirichlet-to-Neumann Maps
			3.1 Abstract Preliminaries
			3.2 Fitting PDEs into the Framework
		4 Applications
		References
	Operator-Valued Nevanlinna–Herglotz Functions, Trace Ideals, and Sergey Naboko\'s Contributions
		1 Introduction
		2 Basic Facts on Bounded Operator-Valued Nevanlinna–Herglotz Functions
		3 Trace Ideals and Sergey Naboko\'s Contributions
		References
	Mathematical Heritage of Sergey Naboko: Functional Models of Non-Self-Adjoint Operators
		1 Dilation Theory for Dissipative Operators
			1.1 Additive Perturbations
		2 Naboko\'s Functional Model for a Family of Additive Perturbations
			2.1 Isometric Map Between the Dilation and Model Spaces
			2.2 Model Representation of Additive Perturbations
			2.3 Smooth Vectors and the Absolutely Continuous Subspace
			2.4 Scattering Theory
			2.5 Singular Spectrum of Non-self-adjoint Operators
			2.6 A Functional Model Based on the Strauss Characteristic Function
			2.7 Applications of the Functional Model Technique
		References
	On Crossroads of Spectral Theory with Sergey Naboko
		1 Embedded Eigenvalues
		2 Magnetohydrodynamics
		3 Operators on Metric Graphs
		References
	Sergey Naboko\'s Legacy on the Spectral Theory of JacobiOperators
		References
	On the Work by Serguei Naboko on the Similarity to Unitaryand Selfadjoint Operators
		1 The Results and a Discussion
		2 Applications
		3 Similarity to Normal Operators
		References
Part II Research Contributions
	Functional Models of Symmetric and Selfadjoint Operators
		1 Introduction
		2 Preliminaries
			2.1 Linear Relations
			2.2 Ordinary Boundary Triples and Weyl Functions
			2.3 B-Generalized Boundary Triples
			2.4 Weyl Function and Spectral Multiplicity
		3 Functional Models in L2(,H)
			3.1 Space L2(,H)
			3.2 The Lebesgue-Stieltjes Integral I with Respect to the Operator Measure
			3.3  Functional Model of a Symmetric Operator in H(M)
			3.4 Functional Models for Proper Extensions of a Symmetric Operator
		4 Functional Models in Reproducing Kernel Hilbert Spaces
			4.1 Reproducing Kernel Hilbert Space L(M)
			4.2 Functional Model in L(M)
			4.3 Unitary Equivalence
		5 Perturbation Theory and Functional Models
			5.1 The Case of Additive Perturbation
			5.2 Applications to de Branges-Rovnyak and Carey Perturbation Results
			5.3 H-1-perturbations of Selfadjoint Operators
		References
	Schrödinger Operators with δ-potentials Supported on Unbounded Lipschitz Hypersurfaces
		1 Introduction
		2 The Schrödinger Operator with δ-potential Supported on a Lipschitz Graph
			2.1 The Form aα and the Operator Aα
			2.2 Essential Spectrum of Aα
			2.3 Uniqueness of the Ground State
		3 The Birman-Schwinger Principle and an Optimization Result for δ-potentials on a Hyperplane
			3.1 The Birman-Schwinger Principle for δ-potentials Supported on a Hyperplane
			3.2 Optimization of λ1(α) and the Symmetric Decreasing Rearrangement
		Appendix A
		References
	Improved Lieb–Thirring Type Inequalities for Non-selfadjoint Schrödinger Operators
		1 Introduction
		2 New Lieb–Thirring Type Inequalities
		3 Examples
		References
	Ballistic Transport in Periodic and Random Media
		1 Introduction
			1.1 Background
			1.2 Contents
		2 The Free Laplacian
			2.1 Discrete Case
			2.2 Continuous Case
		3 Periodic Operators in Euclidean Space
			3.1 Continuous Case
			3.2 Periodic Discrete Graphs
				3.2.1 Preliminaries
				3.2.2 The Schrödinger Operator
		4 Universal Covers
			4.1 Background
			4.2 Universal Covers
			4.3 The Anderson Model on Universal Covers
		5 Epilogue: Limiting Distributions
			5.1 General Facts
			5.2 Periodic Models
		Appendix A: Upper Bounds and Derivatives
			A.1 Lower Bounds
			A.2 Discrete Case
			A.3 Continuous Case
		References
	On the Spectral Theory of Systems of First Order Equations with Periodic Distributional Coefficients
		1 Introduction
		2 Basic Properties of Periodic Distributions
		3 Floquet Theory
		4 Spectral Theory
		5 The Case n=1
		6 The Case n=2, Real Coefficients
		7 Examples
		References
	Asymptotic Analysis of Operator Families and Applications to Resonant Media
		1 Introduction
		2 Functional Models for Dissipative and Nonselfadjoint Operators
			2.1 Lax-Phillips Theory
				2.1.1 Minimality, Non-selfadjointness, Resolvent
			2.2 Pavlov\'s Functional Model and Its Spectral Form
				2.2.1 Additive Perturbations 12:MR0365199, 12:MR0510053
				2.2.2 Extensions of Symmetric Operators 12:Drogobych
				2.2.3 Pavlov\'s Symmetric Form of the Dilation
				2.2.4 Naboko\'s Functional Model of Non-selfadjoint Operators
			2.3 Functional Model for a Family of Extensions of a Symmetric Operator
				2.3.1 Boundary Triples
				2.3.2 Characteristic Functions
				2.3.3 Functional Model for a Family of Extensions
				2.3.4 Smooth Vectors and the Absolutely Continuous Subspace
				2.3.5 Wave and Scattering Operators
				2.3.6 Spectral Representation for the Absolutely Continuous Part of the Operator A0 and the Scattering Matrix
			2.4 Functional Models for Operators of Boundary Value Problems
				2.4.1 Boundary Value Problem
				2.4.2 Family of Boundary Value Problems
				2.4.3 Functional Model
			2.5 Generalised Resolvents
			2.6 Universality of the Model Construction
				2.6.1 Characteristic Function of a Linear Operator 12:Strauss1960
				2.6.2 Examples
		3 An Application: Inverse Scattering Problem for Quantum Graphs
		4 Zero-Range Potentials with Internal Structure
			4.1 Zero-Range Models
			4.2 Connections with Inhomogeneous Media
			4.3 A PDE Model: BVPs with a Large Coupling
				4.3.1 Problem Setup
				4.3.2 Norm-Resolvent Convergence to a Zero-Range Model with an Internal Structure
				4.3.3 Internal Structure with Higher Dimensions of the Internal Space E
			4.4 The Rôle of Generalised Resolvents
		5 Applications to Continuum Mechanics and Wave Propagation
			5.1 Scaling Regimes for High-Contrast Setups
			5.2 Homogenisation of Composite Media with Resonant Components
				5.2.1 Physical Motivation
				5.2.2 Operator-Theoretic Motivation
				5.2.3 Prototype Problem Setups in the PDE Context
				5.2.4 Gelfand Transform and Direct Integral
				5.2.5 Homogenised Operators and Convergence Estimates
		References
	On the Number and Sums of Eigenvalues of Schrödinger-type Operators with Degenerate Kinetic Energy
		1 Introduction
		2 Preliminaries
			2.1 Trace Ideals
			2.2 Fourier Restriction and Extension
		3 Bounds on Number and Sums of Functions of Eigenvalues in L2(Rd)
			3.1 Number of Eigenvalues below a Threshold
			3.2 Sums of Powers of Eigenvalues
			3.3 Sums of Logarithms of Eigenvalues
			3.4 CLR Bounds in L2(Rd)
		4 Schrödinger Operators with Degenerate Kinetic Energy in 2(Zd)
			4.1 Laplace and BCS-Type Operators in 2(Zd)
				4.1.1 Ordinary Lattice Laplace
				4.1.2 Molchanov–Vainberg Laplace
				4.1.3 An Analog of the BCS Operator in 2(Zd)
			4.2 Number of Eigenvalues below a Threshold
			4.3 Sums of Powers of Eigenvalues
			4.4 Sums of Logarithms of Eigenvalues
			4.5 A CLR Bound for Powers of the BCS Operator in 2(Zd)
		5 Alternative Proof of Theorem 3.2
		References
	Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators
		1 Introduction
			1.1 Setting
			1.2 Examples
			1.3 Deriving or Preventing Cantor Spectrum
			1.4 Gap Labelling via K-Theory
			1.5 Organization
		2 Oscillation Theory
		3 The Integrated Density of States
		4 Flows, Suspensions, and the Schwartzman Homomorphism
			4.1 Basics
			4.2 The Suspension of a Dynamical System
			4.3 The Schwartzman Homomorphism
		5 The Gap Labelling Theorem
		6 Almost-Periodic Potentials
			6.1 Examples
			6.2 Generalities about Almost-Periodic Sequences
			6.3 The Frequency Module
		7 Subshift Potentials
			7.1 Schwartzman Group Associated with a General Subshift
			7.2 Subshifts Generated by Substitutions
			7.3 Full Shift over a Finite Alphabet
		8 Potentials Generated by Affine Torus Homeomorphisms
		References
	Degenerate Elliptic Operators and Kato\'s Inequality
		1 Introduction
		2 Maximal and Minimal Operators
		3 Density
		4 Semigroup Generation
		5 The Kato Inequality
		References
	Generalized Indefinite Strings with Purely Discrete Spectrum
		1 Introduction
		Notation
		2 Generalized Indefinite Strings
		3 Some Integral Operators in 10[0,L)
		4 Quadratic Operator Pencils
		5 Purely Discrete Spectrum
		6 The Isospectral Problem for the Conservative Camassa–Holm Flow
		7 Schrödinger Operators with δ\'-interactions
		Appendix A: On a Class of Integral Operators
		Appendix B: Linear Relations
		References
	Soliton Asymptotics for the KdV Shock Problem of Low Regularity
		1 Introduction and Main Results
		2 From the Initial RHP to the Pre-model RHP
			2.1 Statement of the Initial RH Problem
			2.2 Properties of the Scattering Data and Their Analytic Continuations
			2.3 Estimates for the Jump Matrices
		3 Solution of the Model Problem and Final Asymptotic Analysis
		References
	Realizations of Meromorphic Functions of Bounded Type
		1 Introduction
		2 Preliminaries
			2.1 Realizations
			2.2 Functions of Bounded Type
				2.2.1 Herglotz-Nevanlinna Functions
				2.2.2 Generalized Nevanlinna Functions
				2.2.3 Extended Nevanlinna Class
		3 Realizations of Möbius Transforms
		4 Quasi-Herglotz Functions
		5 Main Theorem
		References
	Spectral Transition Model with the General Contact Interaction
		1 Introduction
		2 The Model
		3 The Quadratic Form
		4 The Jacobi Operator
		5 Self-adjointness of the Hamiltonian
		6 Absolutely Continuous Spectrum
		7 Discrete Spectrum
		8 Appendix: Asymptotics of Solutions of the Jacobi Equation
		References
	Weyl\'s Law under Minimal Assumptions
		1 Introduction and Main Result
		2 The Case γ=1
			2.1 Basic Properties of Coherent States
			2.2 Lower Bound on `3́9`42`\"̇613A``45`47`\"603ATr(-h2Ω+V)-
			2.3 Upper Bound on `3́9`42`\"̇613A``45`47`\"603ATr(-h2Ω+V)-
		3 The Case γ>1
		4 The Case γ<1
			4.1 A Weak Convergence Result
			4.2 Lower Bound on `3́9`42`\"̇613A``45`47`\"603ATr(-h2Ω+V)-γ
			4.3 Upper Bound on `3́9`42`\"̇613A``45`47`\"603ATr(-h2Ω+V)-γ
		5 Extension to the Magnetic Case
			5.1 The Upper Bound on `3́9`42`\"̇613A``45`47`\"603ATr( Mh + V )-γ for γ≥3/2
			5.2 The Lower Bound on `3́9`42`\"̇613A``45`47`\"603ATr( Mh + V )-γ for γ≥1
				5.2.1 The Case γ=1
				5.2.2 The Case γ>1
			5.3 Proof of Theorem 5.1
		References
	Weyl–Titchmarsh M-Functions for φ-Periodic Sturm–Liouville Operators in Terms of Green\'s Functions
		1 Introduction
		2 Some Background for Sturm–Liouville Differential Operators
		3 The φ-Periodic Green\'s Function and Elements of Floquet Theory
		4 The Weyl–Titchmarsh Function in the φ-Periodic Case in Terms of the Green\'s Function and Its Nevanlinna–Herglotz Property
		Appendix A: Classical Weyl–Titchmarsh Theory in the Case of Separated Boundary Conditions
		References
	On Discrete Spectra of Bergman–Toeplitz Operators with Harmonic Symbols
		1 Introduction
		2 Some Preliminaries
			2.1 Generalities from Operator Theory
			2.2 Reminder on Hilbert-Schmidt Operators
			2.3 On the Discrete Spectrum of a Perturbed Operator: A Result of Favorov–Golinskii
		3 Proof of the Main Result
		References
	One Dimensional Discrete Schrödinger Operators with Resonant Embedded Eigenvalues
		1 Introduction
		2 Some Basic Lemmas
		3 Technical Preparations
		4 Constructions of Potentials and Proof of Theorems 1.1 and 1.2
		References
	On the Invariance Principle for a Characteristic Function
		1 Introduction
		2 Preliminaries and Basic Definitions
		3 A Functional Model of a Triple
		4 The Invariance Principle
		5 Invariance Principe for Model Triples
		6 Proof of Theorem 4.2
		7 Applications to the Krein-von Neumann Extensions Theory
		References
	A Trace Formula and Classical Solutions to the KdV Equation
		1 Introduction
		2 Notations
		3 Our Framework and Main Ingredients
			3.1 Scattering Data
			3.2 An Oscillatory Integral
		4 Prove of the Main Theorem
		References
	Semiclassical Analysis in the Limit Circle Case
		1 Introduction
			1.1 Setting the Problem
			1.2 Limit Point versus Limit Circle
			1.3 Plan of the Paper
		2 The Semiclassical Ansatz
			2.1 Regular Solutions
			2.2 Jost Solutions
			2.3 Arbitrary Solutions of the Homogeneous Equation
			2.4 Conditions on the Coefficients
		3 Schrödinger Operators and Their Quasiresolvents
			3.1 Minimal and Maximal Operators
			3.2 Quasiresolvent of the Maximal Operator
		4 Self-adjoint Extensions and Their Resolvents
			4.1 Domains of Maximal Operators
			4.2 Self-adjoint Extensions
			4.3 Resolvent
			4.4 Spectral Measure
			4.5 Concluding Remarks
		References