## Course: Functional Analysis II (Winter 2017-2018)Prof. Phan Thanh Nam and Dinh-Thi Nguyen## Unofficial lecture notes by Martin Peev## Homework Sheets## General InformationGoal: We study the spectral theory, with applications to Schroedinger operators and quantum physics.Audience: Master students of Mathematics and Physics, TMP-Master. ## Time and place:- Lectures: Tuesday and Thursday, 12:15-14:00, A027.
- Exercises: Friday, 12:15-14:00, C113.
- G. Teschl, Mathematical methods in quantum mechanics, AMS 2009.
- Reed
and Simon, Methods of modern mathematical physics, Volume I (Functional
analysis, 1980) and Volume IV (Analysis of operators, 1978).
Office hours: Thursday 14:00-15:00 (P.T. Nam, room 328) and Friday 14:00-15:00 (D.T. Nguyen, room 334). Grade: Final grade is determined by your performance on final exam, midterm exam and homework exercises. - You can get up to 100 points from the final exam.
- You can get a bonus of 10 points if you solve at least half of problems in midterm exam.
- You can get a bonus of 10 points if you solve at least half of problems in homework sheets.
Exams: You can bring your notes, including the homework sheets and their solutions. Electronic devices are not allowed. Please remember bring your identity and your student card. ## Brief contents of lectures17/10: Chapter 1: Hilbert spaces and operators. Hilbert spaces. Riesz representation theorem. Orthogonal decomposition. Parseval identity. Orthonormal basis. Separable Hilbert spaces. Weak convergence. Banach-Alaoglu Theorem. Densely defined unbounded operators. Adjoint of an operator. Symmetric operators. Bounded operators. Compact operators.19/10. Spectral Theorem for compact, symmetric operators. Schatten spaces. Trace class operators. 24/10. Hilbert-Schmidt operators. Kernel of an operator. Chapter 2: Self-adjointness. Self-adjoint operators. Motivations: spectral theorem and quantum mechanics. Examples of self-adjoint extensions. 26/10. Operators bounded from below. Quadratic form and Friedrichs' extension. Min-max principle for singular values. 02/11. Kato-Rellich Theorem. Self-adjointness and range of operator. Inverse of perturbations of identity. 07/11. Chaper 3: Spectrum. Resolvent and spectrum. Discrete spectrum and essential spectrum. Closedness of spectrum. Self-adjointness and spectrum. Semi-boundedness and spectrum. Spectrum of multiplication operators. 09/11. Chapter 4: Spectral theorem. Three versions: Multiplication opeartors, Functional calculus, Projection-valued measures. Proof of spectral theorem for bounded self-adjoint operators. 14/11. Proof of spectral theorem for unbounded self-adjoint operator. Applications of the spectral theorem. 16/11. Weyl's criterion for spectrum, essential spectrum and discrete spectrum. Essential spectrum with compact pertubations. 21/11. Chapter 5: Free Schroedinger operator. 23/11. |