Self-Adjoint Operators and Partial Differential Equations
Authors
Description
This book has been written to serve as a text on Functional Analysis, and is by no means intended to be an exhaustive introduction. The author’s aim is to present a representative subset of Functional Analysis in an elementary way and with few prerequisites, in order to highlight the relationship that exists between the concrete problems of Analysis and the abstract formulations of Functional Analysis. The concrete problems considered by the author are some equations and partial derivatives and the corresponding concrete objects are symmetric linear differential operators, and the abstract objects used to study these operators are the self-adjoint operators.
Target audience
Higher education
Name: Self-Adjoint Operators and Partial Differential Equations
Author(s): e Javier Thayer
Pages: 248
Publication: IMPA, 2016
ISBN: 978-85-244-0034-6
Edition: 2
Foreword
CHAPTER I – INTRODUCTION TO FUNCTIONAL ANALYSIS
1. Spaces with inner product
2. Normed vector spaces
3. Continuous linear applications
CHAPTER II – GEOMETRY OF HILBERT SPACES
4. Sequences and orthonormal bases
5. Fourier series
6. Orthogonal projection
7. Riesz representation theorem
8. Direct sums
9. Measurability and integration
10. The adjoint of a bounded operator
11. Multiplication and composition operators
12. Order structure for self-adjoint operators
13. Hilbert-Schmidt operators
14. Symmetric compact operators
CHAPTER III – THE SPECTRAL THEOREM FOR LINEAR OPERATORS
15. Statement of the Spectral Theorem
16. Algebras and representations
17. Spectral homorphisms
18. Functional calculus
19. Demonstration of the Spectral Theorem
CHAPTER IV – THE SPECTRAL THEOREM FOR UNBOUNDED OPERATORS
20. Unbounded operators
21. Symmetric and self-adjoint operators
22. Spectral theorem for self-adjoint operators
23. Extensions of a symmetric operator
24. Functional calculus for self-adjoint operators
25. Spectrum
26. Essential spectrum
27. Friedrichs extension
28. Stone’s theorem
CHAPTER V – INTRODUCTION TO DIFFERENTIAL OPERATORS
29. Generalized functions
30. Differential operators
31. Dirichlet and Neumann extensions
CHAPTER VI – FOURIER TRANSFORM
32. Fourier transform
33. Differential operators with constant coefficients
34. The Laplacian in Rn
CHAPTER VII – THE SCALE OF A SELF-ADJOINT OPERATOR
35. Scaling a self-adjoint operator
36. Interpolation
37. Sobolev spaces
38. Seminormed spaces
CHAPTER VIII – NOTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
39. Eliticity in Rn
40. Local elasticity
41. Heat conduction equation
42. Formalism of quantum mechanics
43. Kato-Rellich theorem
44. Hamiltonian of a particle
CHAPTER IX – MULTIPLICITY THEORY
45. Measurable families of Hilbert spaces
46. Direct integrals
47. Classification of spectral homomorphisms
48. Classification of self-adjoint operators
References