## Course: Functional Analysis II (Winter 2017-2018)Prof. Phan Thanh Nam and Dinh-Thi NguyenNews: You can review your results of the final exam on Friday, February 16, 2018, from 12:00-14:00, room B132. Final exam will take place on Tuesday, February 13, 2018, from 8:30 to 11:30, room B004. You can review your results of the midterm exam in the exercise section on Friday, December 22. Midterm exam will take place on December 19, 2017, from 12:00 to 14:00, room A027. ## 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).
- Lieb-Loss: Analysis, Amer. Math. Soc. 2001.
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 lectures17/10/2017: 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. Self-adjointness and range of operator. Inverse of perturbations of identity. Kato-Rellich Theorem. 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. Self-adjoint extension of Laplacian. Fourier transform and Sobolev spaces. Weak derivatives. Riesz-Thorrin interpolation inequality. Hausdorff-Young inequality. Convolution inequalities. 23/11. Approximation of delta function by convolution. Fundamental lemma of calculus of variations. Hardy-Littlewood-Sobolev inequality. Fourier transform of 1/|x|^s. Sobolev inequality. 28/11. Relativistic kinetic operator. Fractional Sobolev inequality. Sobolev embedding theorem for H^1. 30/11. Heat kernel and Green function of Laplacian. Sobolev embedding theorem for H^s. Newton theorem. 5/12. Chapter 6: Schroedinger operator. Boundedness from below. Self-adjointness on H^2 by Kato-Rellich and essential spectrum. 7/12. Self-adjointness by Friedrichs extension and essential spectrum. Singular potentials 1/|x|^s. Hardy inequality. 12/12. Existence of negative eigenvalues/bound states. Trapping potentials and absence of essential spectrum. 14/12. Chapter 7: Semiclassical estimates. Semiclassical approximation. Lieb-Thirring inequality on the sum of negative eigenvalues. 21/12. Kinetic version of the Lieb-Thirring inequality and Rumin's proof. Cwikel-Lieb-Rozenblum (CLR) inequality on the number of negative eigenvalues. 09/01/2018. Coherent states and resolutions of identity. Weyl asymptotic formula: lower bound. 11/01. Min-max principe for sums of eigenvalues. Weyl asymptotic formula: upper bound. Lieb-Thirring conjecture. Dirichlet Laplacian on a bounded domain. Pólya's conjecture. 16/01. Berezin-Li-Yau inequality. Asymptotics of the sum of Dirichlet eigenvalues. Tauberian lemma. Weyl's law on the distribution of Dirichlet eigenvalues. 18/01. Chapter 8: Many-body Schroedinger operator. Many-body Hamiltonians and Hilbert spaces. Identical particles: bosons and fermions. Kato theorem on the self-adjointness. HVZ theorem on the essential spectrum of N-body Hamiltonian with positive interactions. Proof of HVZ theorem: first inclusion. 23/01. IMS localization formula. Proof of HVZ theorem: second inclusion. Corollary: positivity of ionization energy implies the existence of bound states. 25/01. Ionization problem. Zhislin theorem on existence. Lieb theorem on non-existence. One-body density matrices. Pauli's exclusion principle. 30/01. Tensor product of Hilbert spaces. A basis of anti-symmetric space of Slater determinants. Proof of Pauli exclusion principle. Ground state energy of non-interacting systems. Kinetic energy estimate. Example of hydrogen-like atoms. 01/02. Thomas-Fermi theory: scaling property, existence and uniqueness of minimizer, and equation of minimizer. 06/02. Hardy–Littlewood maximal function: weak and strong type estimates. Fefferman-de la Llave decomposition of Coulomb potential. Lieb-Oxford inequality for Coulomb energy. 08/02. Validity of Thomas-Fermi theory for large atoms. |