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Functional Analysis II (Winter 2020-2021)

Phan Thanh Nam

Lecture notes - Homework - HW discussion - Uni2work - Piazza Forum

News:

The exam will take place on February 12, 2021 (Friday), 9:00-13:00.
Zoom link (Meeting ID: 956 5869 0566, Passcode: 777444). Final exam and Solutions

General Information

Goal: We study the spectral theory of Schrödinger operators, focusing on the semiclassical approximation and applications in large quantum systems.

Audience : Master students of Mathematics and Physics, TMP-Master. Bachelor students will get "Schein" if pass the course.

Time and place:
  • Lectures: Monday and Friday 10:15-12:00. Zoom link (Meeting ID: 956 5869 0566, Passcode: 777444).
  • Exercise: Thursday 8:30-10:00. Zoom link (Meeting ID: 923 3010 6132, Passcode: 777444)

References:
  • Lieb-Loss: Analysis, Amer. Math. Soc. 2001.
  • Reed-Simon: Methods of modern mathematical physics, Volume IV (Analysis of Operators, 1978).
  • Simon: Trace Ideals and Their Applications (Second edition), AMS 2005.
  • Lieb-Seiringer: The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2009.

Exercises: The weekly homework and extra exercises will be discussed on the exercise section. 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.

Exam: There will be a written final exam. You can use your notes, including the homework sheets and their solutions.

Preliminary contents of the course

  • Basic spectral properties of Schrödinger operators.
  • Semiclassical approximation and Weyl's law.
  • Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities.
  • Dirichlet Laplacian on bounded domains.
  • Can we hear the shape of a drum?
  • Pólya conjecture and Berezin-Li-Yau inequality.
  • Many-body quantum systems.
  • Thomas-Fermi approximation.
  • Hartree-Fock approximation.
  • Correlation energy of Fermi gases.

Contents of the lectures

2.11.2020. Chapter 1: Introduction.

6.11.2020. Chapter 2: Basic spectral properties of Schrödinger operators. Hilbert spaces. Weak convergence. Self-adjoint operators. Spectral theorem. Kato-Rellich theorem. Friedrichs' extension. Min-max principle. Sobolev's inequality.

9.11.2020. Sobolev's compactness embedding. The compactness of operator f(x)g(p). Schrodinger operators with potentials vanishing at infinity: self-adjointness and essential spectrum. Existence of infinitely bound states for negative long range potentials. Schrodinger operators with trapping potentials.

13.11.2020. Chapter 3: Semiclassical estimates. Cwikel-Lieb-Rozenblum inequality. Replacement of CLR bound in one and two-dimensions. Lieb-Thirring inequality. Kinetic inequality for orthonormal functions.

16.11.2020. Birman-Schwinger Principle and applications. Kato-Seiler-Simon inequality for f(x)g(p).

20.11.2020. Weak Lp norm and weak Schatten norm. Cwikel's inequality and its implication to CLR bound. Chapter 4: Weyl's law. Coherent states. Resolution of identity.

23.11.2020. Weyl's law for the sum of negative eigenvalues of Schrödinger operators.

27.11.2020. Chapter 5: Dirichlet Laplacian. Berezin-Li-Yau inequality. Asymptotics for the sum of eigenvalues. Tauberian lemma. Weyl's law for the distribution of eigenvalues. Pólya's conjecture. Proof of Pólya's conjecture for cubes.

30.11.2020. Proof of Pólya's conjecture for tiling domains. Packing problem. Weyl's conjecture for the second order expansion of large eigenvalues. Proof of Weyl's conjecture for cubes. Gauss circle problem. Proof of Sierpiński's estimate via Fourier transform.

4.12.2020. Can one hear the shape of a drum? A 2D counterexample by transplantation method. Chapter 6: Neumann Laplacian. Essential spectrum vs. Compact resolvent.

7.12.2020. Smooth domains and Extension theorem. Kröger's upper bound for Neumann eigenvalues.

11.12.2020. Lieb-Thirring inequality for Neumann Laplacian. Weyl's law for Neumann eigenvalues.

14.12.2020. Pólya's conjecture for Neumann eigenvalues. Proof of Pólya's conjecture for tiling domains. Chapter 7: Many-body quantum systems. Slater determinants. Reduced density matrices.

18.12.2020. Pauli's exclusion principle. Coleman's theorem for admissible one-body density matrices. Two-body density matrices and N-representability problem. Ground state energy of ideal gas. Example of hydrogen-like atom.

21.12.2020. Chapter 8: Thomas-Fermi theory. Density functional theory. Representability of one-body density. Levy–Lieb theory. Thomas-Fermi theory. Convergence of kinetic density functional.

8.1.2021. Convergence of the Levy–Lieb to Thormas-Fermi functional. Convergence of ground state energy and ground states. Atomic Thomas-Fermi minimizer.

11.1.2021. Chapter 9: Hartree-Fock theory. Restriction to Slater determinant. Lieb's variational principle. Hardy-Littewood maximal function. Lieb-Oxford inequality.

14.1.2021. Error bound for atomic reduced Hartree-Fock theory. Scott's correction.

18.1.2021. Bach's correlation inequality. Error bound for atomic Hartree-Fock theory. Hartree-Fock energy of a homogeneous Fermi gas: error bound.

22.1.2021. Hartree-Fock energy of a homogeneous Fermi gas: minnimizer is plane waves. Chapter 10: Correlation energy. Fock space formalism. Creation and annihilation operators. Second quantization of one- and two-body operators.

25.1.2021. Particle–hole transformation. Estimates for kinetic and number operators. Removing the non-bosonizable term.

29.1.2021. Bogoliubov diagonalization of bosonizable term. Conclusion of the derivation of the correlation energy. Chapter 11: Stability of matter. Teller’s no-binding theorem for Thomas-Fermi theory.

1.2.2021. First proof of the stability of matter. Baxter’s electrostatic inequality.

5.2.2021. Second proof of the stability of matter. Existence of thermodynamic limit. Grand-canonical stability.

8.2.2021. Instability of bosons. N^{5/3} law and N^{7/5} law.