Advanced Topics in Mathematical Quantum Theory

SUMMARY OF LECTURES

Lecture 1. Perturbations of Schroedinger operators. Discrete and essential spectrum. Weyl's theorem.

Lecture 2. Example: The harmonic oscillator. Creation and annihilation operators. Spectrum, eigenfunctions. Properties of the eigenfunctions.

Lecture 3. Example: the finite well potential. Spectrum, eigenstates, generalized eigenstates.

Lecture 4. Variational formulation of the Schroedinger equation. Energy functional. Sobolev spaces, Sobolev inequalities. Lower bound for the energy functional.

Lecture 5. Continuity of the potential energy. Properties of Sobolev spaces, weak convergence implies strong convergence on small sets.

Lecture 6. Variational interpretation of solutions of the time-independent Schroedinger equation. Existence of ground state for the energy functional. Existence of excited states.

Lecture 7. Min-max principles, and its generalizations. Bounds on sums of eigenvalues. The Dirichlet Laplacian. Spectrum.

Lecture 8. Bathtub principle, Li-Yau inequality.

Lecture 9. Asymptotic behavior of eigenvalues: Dirichlet Laplacian in a box. Semiclassical approximation. Weyl law.

Lecture 10. Coherent states and their properties.

Lecture 11. Upper bound on the sum of eigenvalues of the Dirichlet Laplacian: proof of the Weyl law.

Lecture 12. Semiclassical approximation for the sum of the negative eigenvalues of Schrodinger operators. Lieb-Thirring inequality. Many-body quantum systems: bosons and fermions.

Lecture 13. Example of antisymmetric state: Slater determinants. Models for atoms and molecules. The problem of stability of matter, of first and second kind. rho^{5/3} approximation of the kinetic energy: the free Fermi gas.

Lecture 14. Thomas-Fermi theory. The TF energy functional and its domain. Interpolation inequalities, Hardy-Littlewood-Sobolev. Outline of the strategy to prove existence of the minimizer.

Lecture 15. Existence of a minimizer for the TF energy functional. Properties of electrostatic energies.

Lecture 16. Newton's theorem. Application: lower bound for the TF energy functional.

Lecture 17. Proof of the uniform lower bound for the TF energy. Convexity of the TF functional. Uniqueness of the minimizer.

Lecture 18. Ionization energy in TF theory. Critical number of particles. The TF equation.

Lecture 19. Proof of nonnegativity of the TF potential, computation of the critical number of particles. The ground state energy of neutral atoms in TF theory.

Lecture 20. The no-binding theorem. Baxter's lemma.

Lecture 21. Proof of the no binding theorem. Stability of matter of the second kind for TF theory. Bounds on the sum of negative eigenvalues: The Lieb-Thirring inequality.

Lecture 22. The Lieb-Thirring kinetic energy inequality. Proof of stability of matter of the second kind for many-body quantum systems. Reduced one-particle density matrix, upper bound for fermions.

Lecture 23. Validity of TF theory for large quantum systems. Upper bound on the many-body ground state energy. Hartree-Fock theory. Lieb's variational principle. Coherent states.

Lecture 24. Conclusion of the upper bound for the many-body energy. Proof of Lieb's variational principle. Lower bound for the many-body energy.

References. 

L. C. Evans, Partial Differential Equations. AMS.

E. H. Lieb. Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603-641 (1981).

E. H. Lieb and M. Loss. Analysis. AMS.

E. H. Lieb and R. Seiringer. Stability of Matter. Cambridge University Press.

M. Reed and B. Simon. Methods of Modern Mathematical Physics. Academic Press.

G. Teschl. Mathematical Methods in Quantum Mechanics. AMS.