Mathematical Analysis and Numerical Methods for Science and Technology

VIII. Spectral Theory.- §1. Elements of Spectral Theory in a Banach Space. Dunford Integral and Functional Calculus.- 1. Resolvant Set and Resolvant Operator. Spectrum of A.- 2. Resolvant Equation and Spectral Radius.- 3. Dunford Integral and Operational Calculus.- 4. Isolated Singularities of the Resolvant.- §2. Spectral Decomposition of Self-Adjoint and Compact Normal Operators in a Separable Hilbert Space and Applications.- 1. Hilbert Sums.- 2. Spectral Decomposition of a Compact Self-Adjoint Operator.- 3. Spectral Decomposition of a Compact Normal Operator.- 4. Solution of the Equations Au = f. Fredholm Alternative.- 5. Examples of Applications.- 6. Spectral Decomposition of an Unbounded Self-Adjoint Operator with Compact Inverse.- 7. Sturm-Liouville Problems and Applications.- 8. Application to the Spectrum of the Laplacian in ???n.- 9. Determining the Eigenvalues of a Self-Adjoint Operator with Compact Inverse. Min-Max and Courant-Fisher Formulas.- §3. Spectral Decomposition of a Bounded or Unbounded Self-Adjoint Operator.- 1. Spectral Family and Resolution of the Identity. Properties.- 2. Spectral Family Associated with a Self-Adjoint Operator; Spectral Theorem.- 3. Properties of the Spectrum of a Self-Adjoint Operator. Multiplicity. Examples.- 4. Functions of a Self-Adjoint Operator.- 5. Operators which Commute with A and Functions of A.- 6. Fractional Powers of a Strictly Positive Self-Adjoint Operator.- § 4. Hubert Sum and Hilbert Integral Associated with the Spectral Decomposition of a Self-Adjoint Operator A in a Separable Hilbert Space H*.- 1. Canonical Representation Associated with a Self-Adjoint Operator Whose Spectrum is Simple.- 2. Hilbert Sum Associated with the Spectral Decomposition of a Self-Adjoint Operator A in a Separable (and Complex) Hilbert Space H.- 3. Hilbert Integral. Diagonalisation Theorem of J. von Neumann and J. Dixmier.- 4. An Application: The Intermediate Derivative and Trace Theorems.- 5. Generalised Eigenvectors.- Appendix. "Krein-Rutman Theorem"*.- IX. Examples in Electromagnetism and Quantum Physics*.- A. Examples in Electromagnetism.- §1. Basic Tools: Gradient, Divergence and Curl Operators.- 1. Introduction. Definitions (Gradient, Divergence, Curl).- 2. The Spaces H (div, ?) and H (curl, ?). Principal Properties.- 3. Kernel and Image of the Gradient, Divergence and Curl Operators. Introduction.- 4. Some Results on Regularity.- §2. Static Electromagnetism.- 1. Magnetostatics of a Surface Current.- 2. Electrostatics of a Surface Charge.- Review of § 2.- §3. The Spectral Problem in a Bounded Open Domain (Cavity) with Perfect Conductor Boundary Conditions.- 1. Definition and Fundamental Properties of the Maxwell Operator A in an Open Domain ? ??3 with Bounded Boundary ? = ??.- 2. Spectral Properties of A in a Bounded Open Domain (Cavity).- Review of § 3.- §4. Spectral Problems in a Wave Guide (Cylinder).- 1. Introduction.- 2. The Maxwell Operator A in a Cylinder. Definition of D (A) and the Trace Theorem.- 3. Study of the Kernel of the Operator A in the Space H.- 4. Spectral Decomposition of the Maxwell Operator A in the Case of a Cylinder ("Wave Guide") ? = ?T × ? with ?T a Connected and Regular, Bounded Open Domain in ?2.- Spaces Utilised.- B. Examples in Quantum Physics.- on the Observables of Quantum Physics.- §1. Operators Corresponding to the Position, Momentum and Angular Momentum Observables.- 1. System Consisting of a Single "Non Relativistic" Particle Without Spin, Located in the Space ?3.- 2. System Consisting of a Single "Non Relativistic" Particle with Spin (1/2) in ?3.- 3. System of a Single Particle Located in a Bounded Domain ???3.- 4. System of N Distinguishable non Relativistic Particles in ?3.- 5. System of N Indistinguishable non Relativistic Particles in ?3.- 6. System of a Single Free Relativistic Particle. Case of a Particle with Spin 1/2 Satisfying the Dirac Equation.- 7. Other Cases of Relativistic Particles.- §2. Hamiltonian Operators in Quantum Physics.- 1. Definition of Hamiltonian Operators as Self-Adjoint Operators.- 2. Hamiltonian Operators and Essentially Self-Adjoint Operators.- 3. Unbounded Below Hamiltonian Operators.- 4. (Discrete) Point Spectrum, and Essential Spectrum of (Hamiltonian) Self-Adjoint Operators.- 5. Continuous Spectrum of (Hamiltonian) Self-Adjoint Operators.- Appendix. Some Spectral Notions.- 2. "Continuous Operational Calculus" for a (Bounded) Normal Operator.- 3. "Continuous Operational Calculus" for an (Unbounded) Self-Adjoint Operator.- 4. Simultaneous Spectrum of a Commutative Family of (Bounded) Normal Operators.- 8. von Neumann Algebras.- 9. "Bounded Operational Calculus".- 10. Maximal Commutative von Neumann Algebras.- 11. "Maximal" Spectral Decomposition. "Complete Family of Observables which Commute".- Table of Notations.- of Volumes 1, 2, 4-6.