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.
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 lectures17/10/2017: 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.