Schrödinger Operators, Spectral Analysis and Number Theory : In Memory of Erik Balslev

This book gives its readers a unique opportunity to get acquainted with new aspects of the fruitful interactions between Analysis, Geometry, Quantum Mechanics and Number Theory. The present book contains a number of contributions by specialists in these areas as an homage to the memory of the mathematician Erik Balslev and, at the same time, advancing a fascinating interdisciplinary area still full of potential. Erik Balslev has made original and important contributions to several areas of Mathematics and its applications. He belongs to the founders of complex scaling, one of the most important methods in the mathematical and physical study of eigenvalues and resonances of Schrödinger operators, which has been very essential in advancing the solution of fundamental problems in Quantum Mechanics and related areas. He was also a pioneer in making available and developing spectral methods in the study of important problems in Analytic Number Theory. Preface Short Biography of Erik Balslev Scientific Contributions of Erik Balslev List of Publications by Erik Balslev Erik, My Life Companion Our Friend, Our Colleague Erik Balslev Contents Introduction to the Scientific Contributions in the Book Asymptotics of Random Resonances Generated by a Point Process of Delta-Interactions 1 Introduction 2 Point Process of Random Resonances 3 Asymptotic Density and Weyl-Type Asymptotics with Probability 1 3.1 Proofs of Theorems 3.1 and 3.2 4 Point Process Describing the Asymptotics of Random Resonances 5 Limits of Random Asymptotic Structures Under Growing Intensity 5.1 Limit Law for the `Most Narrow' Asymptotic Sequence 5.2 Estimates on the Growth of the Total Asymptotic Density References Green's Functions and Euler's Formula for ζ(2n) 1 Introduction 2 Bernoulli Polynomials and Bernoulli Numbers 3 On Trace Class Operators 4 The Green's Function for (- ΔD)n, n inmathbbN 5 Some Generalizations References On Courant's Nodal Domain Property for Linear Combinations of Eigenfunctions Part II 1 Introduction 1.1 Notation 1.2 Courant's Nodal Domain Theorem 2 The Equilateral Rhombus 2.1 Symmetries and Spectra 2.2 Riemann-Schwarz Reflection Principle 2.3 Some Useful Results 2.4 Rhombus with Neumann Boundary Condition 2.5 ECP(mathcalRhe,mathfrakn) Is False 2.6 Numerical Results for the ECP(mathcalRhe,mathfrakn) 2.7 Numerical Results for the ECP(mathcalRhe,mathfrakd) 3 The Regular Hexagon 3.1 Symmetries and Spectra 3.2 Symmetries and Boundary Conditions on Sub-domains 3.3 Identification of the First Dirichlet Eigenvalues of the Regular Hexagon 3.4 Numerical Results and ECP(mathcalH,mathfrakd) 3.5 Identification of the First Neumann Eigenvalues of the Regular Hexagon 3.6 Numerical Computations and ECP(mathcalH,mathfrakn) 4 Final Comments 4.1 Numerical Computations 4.2 Final Remarks References Asymptotic Behavior of Large Eigenvalues of the Two-Photon Rabi Model 1 Introduction 1.1 The Two-Photon Rabi Model 1.2 Main Result 1.3 Plan of the Paper 1.4 Notations and Conventions 2 Formulation in Terms of Jacobi Matrices 2.1 Jacobi Matrices 2.2 Jacobi Matrices and the Two-Photon Hamiltonian 3 A New Family of Jacobi Matrices 3.1 Jacobi Matrices Jγδ 3.2 Proof of Lemma 3.2 3.3 Proof of Proposition 2.3 in the Case Δ=0 4 Extension to ell2(mathbbZ) 4.1 Preliminaries 4.2 A Unitary Change of Variable 4.3 Proof of Theorem 4.1 5 Eigenvalue Asymptotics for Lγδ 5.1 Auxiliary Results 5.2 Proof of Proposition 5.1 5.3 About the Asymptotics Conjecture 6 End of the Proof of Theorem 1.1 6.1 Proof of Proposition 3.1 6.2 Proof of (1.5) References Some Remarks on Spectral Averaging and the Local Density of States for Random Schrödinger Operators on L2 (mathbbRd) 1 Statement of the Problem and Result 1.1 Contents 2 Trace Estimates from the Poincaré Inequality 3 An Alternate Approach to Spectral Averaging 4 Lipschitz Continuity of the Local DOS References Resonances in the One Dimensional Stark Effect in the Limit of Small Field 1 Introduction 2 Resonance—Free Regions 3 Vanishing of the Reflection Coefficient 4 Resolvent Convergence 5 Resonances Near the Negative Real Axis 6 Resonances Near the Positive Real Axis 7 Resonances Near the Line argk = -π/3 8 Higher Dimensions References On the Spectral Gap for Networks of Beams 1 Introduction 2 The Operator 3 On Spectral Properties of Ast (Γ). 4 Surgery for Standard Beam Operators 5 Estimates for the Spectral Gap References Some Notes in the Context of Binocular Space Perception 1 Introduction 2 Biophysical Background 3 Exploring Binocular Vision in Reduced Cue Conditions 4 A Closer Look at the FPL 5 Acquisition of Visual Information 6 Conclusion References Symbolic Calculus for Singular Curve Operators 1 Introduction 2 Hilbert Spaces Associated to New Quantizations of the Sphere 2.1 The Standard Geometric Quantization of the Sphere 2.2 The Building Vectors 2.3 The Hilbert Structure 3 Singular Quantization 3.1 A Toy Model Case 3.2 (General) Trigonometric Matrices 3.3 Action of a General Matrix 3.4 Symbol 3.5 Symbolic Calculus 3.6 A-Töplitz Quantization 3.7 Classical Limit and Underlying ``Phase-Space'' 4 Application to TQFT 4.1 The Curve Operators in the Case of the Once Punctured Torus or the 4-Times Punctured Sphere 4.2 Main Result 4.3 Proof of Theorem 20 4.4 Examples 5 A Conjecture References Higher Order Deformations of Hyperbolic Spectra 1 Introduction 1.1 Erik Balslev's Interest in Spectral Deformations in the Context of Hyperbolic Surfaces 1.2 Higher Order Deformation 2 Stability of Eigenvalues Under Character Deformations 2.1 Standard Non-hololorphic Eisenstein Series 2.2 Character Deformations 2.3 Goldfeld Eisenstein Series 2.4 Higher Order Fermi's Golden Rules 3 Relation to Special Values of Dirichlet Series 4 Idea of Proof References On the Generalized Li’s Criterion Equivalent to the Riemann Hypothesis and Its First Applications 1 Introduction 2 Inequalities Involving the Derivatives of the Logarithm of Riemann ς-Function Related to the Riemann Hypothesis 3 For Any n, the Sums and Derivatives in Question Are Indeed Positive in the Limit of Large b 4 Conclusion Appendix 1: Generating Function for the Generalized Li’s Sums Appendix 2: Asymptotic of Generalized Li’s Sums Assuming the Riemann Hypothesis References Regularising Infinite Products by the Asymptotics of Finite Products 1 Introduction 2 Background and Setting 3 One Dimensional Case 4 Quadratic Two Dimensional Case References Trace Maps Under Weak Regularity Assumptions 1 Introduction 2 Domains with Star Shaped Boundary 3 The Case of Rn 4 The Coarea Formula References