Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications

The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. The objective of the present work is to compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form.

Table of Contents:
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.