This book introduces and discusses the self-adjoint extension problem for symmetric operators on Hilbert space. It presents the classical von Neumann and Krein–Vishik–Birman extension schemes both in their modern form and from a historical perspective, and provides a detailed analysis of a range of applications beyond the standard pedagogical examples (the latter are indexed in a final appendix for the reader’s convenience).
Self-adjointness of operators on Hilbert space representing quantum observables, in particular quantum Hamiltonians, is required to ensure real-valued energy levels, unitary evolution and, more generally, a self-consistent theory. Physical heuristics often produce candidate Hamiltonians that are only symmetric: their extension to suitably larger domains of self-adjointness, when possible, amounts to declaring additional physical states the operator must act on in order to have a consistent physics, and distinct self-adjoint extensions describe different physics. Realising observables self-adjointly is the first fundamental problem of quantum-mechanical modelling.
The discussed applications concern models of topical relevance in modern mathematical physics currently receiving new or renewed interest, in particular from the point of view of classifying self-adjoint realisations of certain Hamiltonians and studying their spectral and scattering properties. The analysis also addresses intermediate technical questions such as characterising the corresponding operator closures and adjoints. Applications include hydrogenoid Hamiltonians, Dirac–Coulomb Hamiltonians, models of geometric quantum confinement and transmission on degenerate Riemannian manifolds of Grushin type, and models of few-body quantum particles with zero-range interaction.
Graduate students and non-expert readers will benefit from a preliminary mathematical chapter collecting all the necessary pre-requisites on symmetric and self-adjoint operators on Hilbert space (including the spectral theorem), and from a further appendix presenting the emergence from physical principles of the requirement of self-adjointness for observables in quantum mechanics.
Author(s): Matteo Gallone , Alessandro Michelangeli
Series: Springer Monographs in Mathematics
Publisher: Springer
Year: 2023
Language: English
Pages: 556
City: Cham
Tags: Hilbert Spaces, Symmetric Operators, Self-Adjoint Operators, Extension Schemes, Hydrogenoid Spectra, Dirac-Coulomb Hamiltonians, Grushin Structures, Zero-Range Interaction
Foreword
Preface
Contents
Part I Theory
1 Generalities on Symmetric and Self-Adjoint Operators on Hilbert Space
1.1 Preliminary Notions
1.2 Bounded, Closed, Closable Operators
1.3 Adjoint Operators
1.4 Minimal and Maximal Realisations of Linear Differential Operators
1.5 Bounded Operators, Compacts, Unitaries, and Orthogonal Projections
1.6 Invariant and Reducing Subspaces
1.7 Spectrum
1.8 Symmetric and Self-Adjoint Operators
1.9 Weyl Limit-Point Limit-Circle Analysis
1.10 Spectral Theorem
1.11 Functional Calculus
1.12 Spectral Theorem in Multiplication Form
1.13 Parts of the Spectrum
1.14 Perturbations of Self-Adjoint Operators
1.15 Quadratic Forms and Self-Adjoint Operators
1.16 Min-Max Principle
2 Classical Self-Adjoint Extension Schemes
2.1 Friedrichs Extension
2.2 Cayley Transform of Symmetric Operators
2.3 von Neumann's Extension Theory of Symmetric Operators
2.4 Kreĭn Transform of Positive Operators
2.5 Kreĭn's Extension Theory of Symmetric Semi-Bounded Operators
2.6 Višik-Birman Parametrisation of Self-Adjoint Extensions
2.6.1 Višik's B Operator
2.6.2 Višik-Birman Extension Representation
2.6.3 Birman's Operator
2.6.4 Semi-Bounded Extensions: Operator Parametrisation
2.6.5 Semi-Bounded Extensions: Quadratic Form Parametrisation
2.6.6 Parametrisation of Friedrichs and Kreĭn-von Neumann Extensions
2.7 Kreĭn-Višik-Birman Self-Adjoint Extension Theory Re-Parametrised
2.8 Invertibility, Semi-Boundedness, and Negative Spectrum in the Kreĭn-Višik-Birman Extension Scheme
2.9 Resolvents in the Kreĭn-Višik-Birman Extension Scheme
2.10 Self-Adjoint Extensions with Friedrichs Lower Bound
Part II Applications
3 Hydrogenoid Spectra with Central Perturbations
3.1 Hydrogenoid Hamiltonians with Point-Like Perturbation at the Centre
3.1.1 Fine Structure and Darwin Correction
3.1.2 Point-Like Perturbations Supported at the Interaction Centre
3.1.3 Angular Decomposition
3.1.4 The Radial Problem
3.1.5 Main Results: Radial Problem, 3D Problem, Eigenvalue Correction
3.2 Hydrogenoid Self-Adjoint Realisations
3.2.1 The Homogeneous Radial Problem
3.2.2 Inhomogeneous Inverse Radial Problem
3.2.3 Distinguished Extension and Its Inverse
3.2.4 Operators ps: [/EMC pdfmark [/Subtype /Span /ActualText (ModifyingAbove upper S With quotation dash) /StPNE pdfmark [/StBMC pdfmarkSps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark, SF, and S*
3.2.5 Kreĭn-Višik-Birman Classification of the Extensions
3.2.6 Reconstruction of the 3D Hydrogenoid Extensions
3.3 Perturbation of the Discrete Spectra
3.3.1 The s-Wave Eigenvalue Problem
3.3.2 Further Remarks
4 Dirac-Coulomb Hamiltonians for Heavy Nuclei
4.1 One-Body Dirac-Coulomb Models in Sub-Critical and Critical Regimes
4.2 Self-Adjoint Realisations of Dirac-Coulomb Blocks of Definite Angular and Spin-Orbit Symmetry
4.3 Extension Mechanism on Each Symmetry Block at Criticality
4.3.1 Deficiency Index Computation
4.3.2 The Homogeneous Problem: Kernel of h*
4.3.3 Distinguished Extension hD
4.3.4 Operator Closure ps: [/EMC pdfmark [/Subtype /Span /ActualText (ModifyingAbove h With quotation dash) /StPNE pdfmark [/StBMC pdfmarkhps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
4.3.5 Resolvents and Spectral Gap
4.4 Sommerfeld Formula and Distinguished Extension
4.4.1 Eigenvalue Problem by Truncation of Asymptotic Series
4.4.2 Eigenvalue Problem by Supersymmetric Methods
4.5 Discrete Spectra for Critical Dirac-Coulomb Hamiltonians
5 Quantum Particle on Grushin Structures
5.1 Grushin-Type Plane and Grushin-Type Cylinder
5.2 Geodesic Incompleteness
5.3 Geometric Quantum Confinement and Transmission Protocols
5.4 Constant-Fibre Direct Integral for the Grushin Plane
5.5 Constant-Fibre Orthogonal Sum for the Grushin Cylinder
5.6 Confinement Mechanisms on Plane and Cylinder
5.7 Related Settings on Almost Riemannian Manifolds
5.8 Extensions on One-Sided Fibre
5.8.1 Homogeneous Differential Problem: Kernel of Aα(k)*
5.8.2 Non-homogeneous Inverse Differential Problem
5.8.3 Operator Closure ps: [/EMC pdfmark [/Subtype /Span /ActualText (ModifyingAbove upper A Subscript alpha Baseline left parenthesis k right parenthesis With quotation dash) /StPNE pdfmark [/StBMC pdfmarkAα(k)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
5.8.4 Distinguished Extension and Induced Classification
5.8.5 One-Sided Extensions for Non-zero Modes
5.8.6 One-Sided Extensions for the Zero Mode
5.9 Extensions on Two-Sided Fibre
5.10 General Extensions of ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper H Subscript alpha) /StPNE pdfmark [/StBMC pdfmarkHαps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
5.11 Uniformly Fibred Extensions of ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper H Subscript alpha) /StPNE pdfmark [/StBMC pdfmarkHαps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
5.11.1 Generalities and Classification Theorem
5.11.2 General Strategy
5.11.3 Integrability and Sobolev Regularity of g0 and g1
5.11.4 Decomposition of the Adjoint Into Singular Terms
5.11.5 Detecting Short-Scale Asymptotics and Regularity
5.11.6 Control of ps: [/EMC pdfmark [/Subtype /Span /ActualText (phi overTilde) /StPNE pdfmark [/StBMC pdfmarkφ"0365φps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
5.11.7 Control of
5.11.8 Proof of the Uniformly Fibred Classification
5.12 Classification of Local Transmission Protocols on Cylinder
5.13 Spectral Analysis of Uniformly Fibred Extensions
5.13.1 Spectral Analysis of the Zero Mode
5.13.2 Spectral Analysis of Non-zero Modes
5.13.3 Reconstruction of the Spectral Content of Fibred Extensions
5.14 Scattering in Transmission Protocols
5.14.1 Scattering on Grushin-Type Cylinder
5.14.2 Scattering on Fibre
6 Models of Zero-Range Interaction for the Bosonic Trimer at Unitarity
6.1 Introduction and Background
6.2 Admissible Hamiltonians
6.2.1 The Minimal Operator
6.2.2 Friedrichs Extension
6.2.3 Adjoint
6.2.4 Deficiency Subspace
6.2.5 Extensions Classification
6.3 Two-Body Short-Scale Singularity
6.3.1 Short-Scale Structure
6.3.2 The Tλ Operator
6.3.3 Large Momentum Asymptotics
6.4 Ter-Martirosyan-Skornyakov Extensions
6.4.1 TMS and BP Asymptotics
6.4.2 Generalities on TMS Extensions
6.4.3 Symmetry and Self-Adjointness of the TMS Parameter
6.4.4 TMS Extensions in Sectors of Definite Angular Momentum
6.5 Sectors of Higher Angular Momenta
6.5.1 Tλ-Estimates
6.5.2 Self-Adjointness for ps: [/EMC pdfmark [/Subtype /Span /ActualText (script l greater than or slanted equals 1) /StPNE pdfmark [/StBMC pdfmark1ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
6.6 Sector of Zero Angular Momentum
6.6.1 Mellin-Like Transformations
6.6.2 Radial Ter-Martirosyan-Skornyakov Equation
6.6.3 Symmetric, Unbounded Below, TMS Extension
6.6.4 Adjoint of the Birman Parameter
6.6.5 Multiplicity of TMS Self-Adjoint Realisations
6.7 The Canonical Model and Other Well-Posed Variants
6.7.1 Canonical Model at Unitarity and at Given Three-Body Parameter
6.7.2 Spectral Analysis and Thomas Collapse
6.7.3 Variants
6.8 Ill-Posed Models
6.8.1 Ill-Posed Boundary Condition
6.8.2 Incomplete Criterion of Self-Adjointness
6.9 Regularised Models
6.9.1 Minlos-Faddeev Regularisation
6.9.2 Regularisation in the Sector =0
6.9.3 High Energy Cut-Off
A Physical Requirements Prescribing Self-Adjointness of Quantum Observables
A.1 Levels of Mathematical Formalisation of Quantum Mechanics
A.2 Physical Requirement for Symmetric Observables: Connection with Unboundedness
A.3 Self-Adjointness of the Quantum Hamiltonian Inferred from the Schrödinger Evolution
A.4 Self-Adjointness of the Quantum Hamiltonian Inferred from the Series Expansion of the Evolution Propagator
A.5 Non-uniqueness of the Schrödinger Dynamics when Self-Adjointness Is Not Declared
A.6 Self-Adjointness of Quantum Observables with an Orthonormal Basis of Eigenvectors
A.7 Self-Adjointness of Quantum Hamiltonians Inferred from Their Physical Stability
B References to Pedagogical Examples
B.1 Quantum Particle on Real Line
B.2 Quantum Particle on Half-Line
B.3 Quantum Particle on Interval
B.4 Quantum Particle with Point Interaction of δ-Type in One Dimension
B.5 Quantum Particle with Point Interaction of δ'-Type in One Dimension
B.6 Quantum Particle with Point Interaction in Two Dimensions
B.7 Quantum Particle with Point Interaction in Three Dimensions
B.8 Quantum Particle with Point Interaction on Domain
B.9 Friedrichs Lower Bound
References
Index
