Content of the lecture
Lecture notes:
J. P. Solovej,
Many
Body Quantum Mechanics, August 30, 2009.
April 16th
Introduction; practical information.
Chapter 1: Preliminaries (up until including 1.17).
Please
sign up
for the Exercise class!
April 18th
Finished Chapter 1.
Chapter 2: The princicples of Quantum Mechanics (up until including 2.1).
April 23th
Continued Chapter 2 (up until including mid page 16).
A note
on a claim in the proof of 2.9.
April 25th
Finished Chapter 2. (The mentioned paper on fractional statistics of
particles in 1 and 2 dimensions).
Finished Chapter 3: Semi-bounded operators and quadratic forms
Chapter 4: Extensions of operators and quadratic forms (up until including 4.2).
April 30th
Finished Chapter 4 (no proof of Thm 4.12; if wanted, discuss in Exercise Class).
Chapter 5: Schroedinger operators (up until including 5.2).
May 2nd
Finished Chapter 5.
May 7th
Chapter 6: The canonical and grand canonical picture and the
Fock spaces (finished).
(The '2Z+1' -
paper by Lieb, and
a 'physics'
-
version (both from 1984, not, as said, 1981). The improvement
for atoms by Nam. For related works, in particular on the
asymptotic regime, see the bibliography of Nam's paper.)
Chapter 7: Second quantization (up until NOT including 7.1).
NB NB The oral exam will take place on July 22nd (Monday). More information soon.
May 14th
Continuation Chapter 7, up until including 7.12 (proof not finished).
May 16th
Finished proof 7.12. Finished Chapter 7.
Chapter 8: One- and two-particle density matrices for bosonic or fermionic states.
Up until including 8.6 (proof not finished).
May 23th
Finished proof 8.6. Continued Chapter 8, up to and including 8.13 (proof 8.12, including Appendix E, starting from next time).
The mentioned papers
by Bardeen-Cooper-Schrieffer
and Yang.
May 28th
Finished Appendix E (including proof of Lemma 8.14), start proof Theorem 8.12 (not finished).
June 4th
Finished proof Theorem 8.12 (including some extra details).
June 6th
Finished Chapter 8.
Chapter
9: Bogolubov transformations.
Up until NOT including formular (52).
June 11th
Up until and including Theorem 9.5. Start proof of 9.5 (one third down on page 57).
June 13th
Continuation of proof of 9.5 (until top of page 59).
June 18th
Finished proof of Theorem 9.5.
June 20th
Proof of necessity of Shale/Shale-Stinespring condition (Appendix F).
June 25th
Continuation in Chapter 9: Lemma 9.6 (including proof) and Theorem 9.9 (start proof; top page 65).
June 27th
Finished proof Theorem 9.9.
Chapter 10: Quasi-free pure states.
Up until and NOT including 10.4.
July 2nd
Finished Chaper 10.
(NOTE: In Theorem 10.4, "alpha alpha^* trace
class" _follows_ from the assumptions on Gamma, just as in Pb. A.8.1
(Homework Sheet 9, Exercise 3)).
Chapter 11: Quadratic Hamiltonians.
Up to and _including_ (proof next time!) Lemma 11.4.
July 4th
Proof of Lemma 11.4.
Up until and including (formulation of) Theorem
11.6. (Proof of Theorem 11.6 next time).
July 9th
Proof of Theorem 11.6.
Chapter 12: Generalised Hartree-Fock Theory.
Finished.
For an example of a study (in generalised Hartree-Fock Theory) of a
system of interacting Fermions with an _attractive_ potential
(gravitation!), see the paper (in particular, last paragraph on p. 260)
"Minimizers
for the Hartree-Fock-Bogoliubov theory of neutron stars
and white dwarfs"
by Enno Lenzmann
and
Mathieu Lewin (arxiv:0809.2560v3).
Chapter 13: Bogolubov Theory.
Up until and including Remark 13.3.
July 11th
Finished Chapter 13 (and hence the Lecture notes!)
July 16th
Discussion of the paper
"Bogoliubov spectrum of interacting Bose gases" (arXiv:1211.2778v3)
by Mathieu
Lewin, Phan Thành Nam,
Sylvia Serfaty, and
Jan Philip Solovej.
July 18th
Finished discussing the paper mentioned under July 16th.
+++++++++++++++++++++++++++++++++++++++++++++++++++
The paper mentioned in the lecture (for a potential seminar) is:
"Bogoliubov spectrum of interacting Bose gases" (arXiv:1211.2778v3)
by Mathieu
Lewin, Phan Thành Nam,
Sylvia Serfaty, and
Jan Philip Solovej.
See also
"Excitation spectrum of interacting bosons in the mean-field infinite-volume limit"
(arXiv:1305.3641v1 )
by Jan Dereziński
and Marcin
Napiórkowski
and
"The Excitation Spectrum for Weakly Interacting Bosons"
(arXiv:1008.5349v2)
by Robert Seiringer.
Other interesting (?!?) papers:
Fermions:
"Translation-invariant quasi-free states for fermionic systems and the BCS approximation"
(arXiv:1305.5135v1)
by Gerhard Bräunlich, Christian Hainzl, Robert Seiringer,
"The BCS Functional for General Pair Interactions" (arXiv:math-ph/0703086v2)
by Christian Hainzl, Eman Hamza, Robert Seiringer, Jan Philip
Solovej,
"Low Density Limit of BCS Theory and Bose-Einstein Condensation of
Fermion Pairs"
(arXiv:1105.1100v1)
by Christian Hainzl and Robert Seiringer,
"On the energy-momentum spectrum of a homogeneous Fermi gas" (arXiv:1111.0787v3)
by Jan Dereziński, Krzysztof A. Meissner, and Marcin Napiórkowski,
Bosons:
"The excitation spectrum for weakly interacting bosons in a trap"
(arXiv:1205.5259v1)
by Philip Grech and Robert Seiringer,
"Derivation of Hartree's theory for generic mean-field Bose systems" (arXiv:1303.0981v3)
by Mathieu Lewin, Phan Thành Nam, and Nicolas Rougerie,
"A Numerical Perspective on Hartree-Fock-Bogoliubov Theory" (arXiv:1206.6081v1)
by Mathieu Lewin and Séverine Paul,
General:
"On the minimization of Hamiltonians over pure Gaussian states" (arXiv:1102.2931v2)
by Jan Dereziński, Marcin Napiórkowski, Jan Philip Solovej,
For other relevant (??) notes on related topics, see for instance
"Geometric methods for nonlinear many-body quantum systems" (arXiv:1009.2836v3)
by Mathieu Lewin,
"Quantum Coulomb Gases" (arXiv:1012.5178v1)
by Jan Philip Solovej
"Bogoliubov theory and bosonic atoms" (arXiv:1109.2875v1)
by Phan Thanh Nam,
"The Energy of Charged Matter" (arXiv:math-ph/0404039v2)
by Jan Philip Solovej,
"The Quantum-mechanical Many Body Problem: The Bose Gas" (arXiv:math-ph/0405004v2)
by Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, and Jakob Yngvason
Disclaimer: The list above is (obvisouly!!) incomplete and biased
(from a certain section of the research community; only
math-papers, no physics papers) etc etc !!!! - Also, may (most?)
of th papers how now appeared in peer reviewed journals; you
should consult the final published version.
Last update: July 18th, 2013 by Thomas Østergaard Sørensen.