## Course: Functional Analysis II (Winter 2017-2018)

Prof. Phan Thanh Nam and Dinh-Thi Nguyen

News:  You can collect your final grade certificate from Mrs. Edith Höchst, office B330.

You can review your results of the final exam on February 16, 2018, 12:00-14:00, room B132.

Final exam will take place on February 13, 2018, 8:30-11:30, room B004.

### 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:
• 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.
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).

• 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

17/10/2017: 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.