Course: Functional Analysis II (Winter 2017-2018)

Prof. Phan Thanh Nam and Dinh-Thi Nguyen

Unofficial lecture notes by Martin Peev

Homework Sheets

General Information

Goal: 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. 
References:
Exercises:  Every week a homework sheet will be posted on Friday. You can handle your solutions in the exercise section on Friday, or put them in the exercise mail box (no. 57, first floor). You are strongly encouraged to do the homework as it is the best way to get familiar with the course's materials and to prepare for the final exam.

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.
You need 50 points to pass and 85 points to get the final grade 1.0.

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 lectures 

17/10: 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.