Course: Functional Analysis II (Winter 2017-2018)Prof. Phan Thanh Nam and Dinh-Thi Nguyen
Unofficial lecture notes by Martin Peev
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:
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.
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.