Functional Analysis and Theory of Operators

Annotation

The functional analysis is extensive. Our course is mainly devoted to the theory of Hilbert spaces, the theory of operators in Hilbert spaces, and applications to quantum mechanics. It will be shown that in quantum mechanics, unbounded operators must arise. The theory of unbounded operators appeared as part of developing a rigorous mathematical framework for quantum mechanics and later became an apparatus for describing models of mathematical physics and a powerful tool for solving problems of mathematical physics. The course introduces students to the basic concepts and techniques connected with the theory of unbounded operators along with the key examples of physically relevant models.

Course plan

1. Hilbert spaces, orthogonal complements, orthonormal basis.
2. Bounded operators and functionals in Hilbert spaces. Adjoint and selfadjoint operators. Spectral theory.
3. Compact operators. Hilbert-Schmidt theorem and Fredholm theory.
4. Fourier transform in L2(R).
5. Unbounded operators. The domain of the operator. The graph of operator.
6. Closed operators, closable operators. Adjoint operator.
7. The deficiency index of operator. The symmetric and self-adjoint operators. Unitary and isometric operators.
8. Self-adjoint extensions of symmetric operators. Von Neumann theory for self-adjoint extensions. Friedrichs extension for nonnegative symmetric operators.
9. Symmetric differential operators, self-adjoint extensions, boundary conditions.
10. Spectral decomposition. Functional calculus of self-adjoint operator.
11. C0-semigroups and infinitesimal generators. Hille–Fillips–Yosida theorem.
12. Functions of noncommuting operators. General notions for noncommutative functional calculus.
13. Sobolev spaces and adjoint to them spaces. Embedding theorems. Fourier transform in Sobolev spaces.
14. Pseudo-differential operators in Soboles spaces. Compositions and adjoint operators for pseudo-differential.
15. Periodic problems. On spectrum of periodic Sturm-Louville operator. Bloch theory.
16. Applications of self-adjoint extension theory for the problems with pointwise potentials.

Literature

1. A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, AMS, 2006, Translations of Mathematical Monographs , vol. 233.
2. 2. M.Reed, B.Simon, Barry, Methods of Modern Mathematical Physics, Volume 1: Functional Analysis (revised and enlarged ed.), (1980), Academic Press.
3. M.Reed, B.Simon, Barry, Methods of Modern Mathematical Physics, Volume 2: Fourier Analysis, Self-Adjointness, 1975, Academic Press.
4. L.Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Springer-Verlag, 2007.
5. M.S.Joshi, Introduction To Pseudo-Differential Operators, arXiv:math/9906155v1.