Mathematics of Bose-Einstein condensation (course webpage)
Where and when: Mondays 13:15-17:00 (room. B.0.17)
Lecture notes: version 2024
Lectures/exercises contents (more or less):
02.10.2024: introduction, motivations, history, many-body Hamiltonian, operators on Hilbert spaces, quantites of interest, tensor products of Hilbert spaces
07.10.2024: Fock spaces, creation and annihilation operators, second quantization of one-body operators, CCR, 2nd quantization of two-body operators, second quantized form of Hamilotnian, spectrum of diagonal operators on Fock space.
14.10.2024: BEC in ideal gas, critical temperature, reduced densities and definition of BEC, ODLRO.
21.10.2024: scaling regimes (mean-field, Gross-Pitaevskii), scattering length
28.10.2024: ground state energy of the mean-field Bose gas
04.11.2024: different ways a sequence of operators can converge, duals of bounded and compact operators, Banach-Alaoglu theorem
18.11.2024: BEC in the ground state of a mean-field Bose gas via quantum de Finetti. Landau criterion for superfliudity
25.11.2024: Bogoliubov approximation. Rigorous formulation of the Bogoliubov approximation
02.12.2024: Sketch of proof of Bogoliubov appoximation
09.12.2024: Sketch of proof of Bogoliubov appoximation cd.
16.12.2024: Elements of abstract theory of quadratic Hamiltonians
09.01.2025: Elements of abstract theory of quadratic Hamiltonians, diagonalization
13.01.2025: Dynamics of bosons, persistence of BEC under Hartree flow
20.01.2025: Coherent states and c-number substitution, Bose gas at positive temperature
Oral exam: Proposed terms: doodle link send via email
Examination topics
Email address: marcin.napiorkowskiINfuw.edu.sk (please replace "IN" by "@" and "sk" by "pl")
October 2024