---
_id: '2370'
abstract:
- lang: eng
text: After recalling briefly the connection between spontaneous symmetry breaking
and off-diagonal long-range order for models of magnets a general proof of spontaneous
breaking of gauge symmetry as a consequence of Bose-Einstein condensation is presented.
The proof is based on a rigorous validation of Bogoliubov's c-number substitution
for the k = 0 mode operator α0.
author:
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jakob
full_name: Yngvason, Jakob
last_name: Yngvason
citation:
ama: Lieb É, Seiringer R, Yngvason J. Bose-Einstein condensation and spontaneous
symmetry breaking. *Reports on Mathematical Physics*. 2007;59(3):389-399.
doi:10.1016/S0034-4877(07)80074-7
apa: Lieb, É., Seiringer, R., & Yngvason, J. (2007). Bose-Einstein condensation
and spontaneous symmetry breaking. *Reports on Mathematical Physics*. Elsevier.
https://doi.org/10.1016/S0034-4877(07)80074-7
chicago: Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “Bose-Einstein Condensation
and Spontaneous Symmetry Breaking.” *Reports on Mathematical Physics*. Elsevier,
2007. https://doi.org/10.1016/S0034-4877(07)80074-7.
ieee: É. Lieb, R. Seiringer, and J. Yngvason, “Bose-Einstein condensation and spontaneous
symmetry breaking,” *Reports on Mathematical Physics*, vol. 59, no. 3. Elsevier,
pp. 389–399, 2007.
ista: Lieb É, Seiringer R, Yngvason J. 2007. Bose-Einstein condensation and spontaneous
symmetry breaking. Reports on Mathematical Physics. 59(3), 389–399.
mla: Lieb, Élliott, et al. “Bose-Einstein Condensation and Spontaneous Symmetry
Breaking.” *Reports on Mathematical Physics*, vol. 59, no. 3, Elsevier, 2007,
pp. 389–99, doi:10.1016/S0034-4877(07)80074-7.
short: É. Lieb, R. Seiringer, J. Yngvason, Reports on Mathematical Physics 59 (2007)
389–399.
date_created: 2018-12-11T11:57:16Z
date_published: 2007-06-01T00:00:00Z
date_updated: 2021-01-12T06:57:04Z
day: '01'
doi: 10.1016/S0034-4877(07)80074-7
extern: 1
intvolume: ' 59'
issue: '3'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0610034
month: '06'
oa: 1
page: 389 - 399
publication: Reports on Mathematical Physics
publication_status: published
publisher: Elsevier
publist_id: '4556'
quality_controlled: 0
status: public
title: Bose-Einstein condensation and spontaneous symmetry breaking
type: journal_article
volume: 59
year: '2007'
...
---
_id: '2371'
abstract:
- lang: eng
text: We give a proof of stability of relativistic matter with magnetic fields all
the way up to the critical value of the nuclear charge Zα = 2/π.
author:
- first_name: Rupert
full_name: Frank, Rupert L
last_name: Frank
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Frank R, Lieb É, Seiringer R. Stability of relativistic matter with magnetic
fields for nuclear charges up to the critical value. *Communications in Mathematical
Physics*. 2007;275(2):479-489. doi:10.1007/s00220-007-0307-2
apa: Frank, R., Lieb, É., & Seiringer, R. (2007). Stability of relativistic
matter with magnetic fields for nuclear charges up to the critical value. *Communications
in Mathematical Physics*. Springer. https://doi.org/10.1007/s00220-007-0307-2
chicago: Frank, Rupert, Élliott Lieb, and Robert Seiringer. “Stability of Relativistic
Matter with Magnetic Fields for Nuclear Charges up to the Critical Value.” *Communications
in Mathematical Physics*. Springer, 2007. https://doi.org/10.1007/s00220-007-0307-2.
ieee: R. Frank, É. Lieb, and R. Seiringer, “Stability of relativistic matter with
magnetic fields for nuclear charges up to the critical value,” *Communications
in Mathematical Physics*, vol. 275, no. 2. Springer, pp. 479–489, 2007.
ista: Frank R, Lieb É, Seiringer R. 2007. Stability of relativistic matter with
magnetic fields for nuclear charges up to the critical value. Communications in
Mathematical Physics. 275(2), 479–489.
mla: Frank, Rupert, et al. “Stability of Relativistic Matter with Magnetic Fields
for Nuclear Charges up to the Critical Value.” *Communications in Mathematical
Physics*, vol. 275, no. 2, Springer, 2007, pp. 479–89, doi:10.1007/s00220-007-0307-2.
short: R. Frank, É. Lieb, R. Seiringer, Communications in Mathematical Physics 275
(2007) 479–489.
date_created: 2018-12-11T11:57:16Z
date_published: 2007-10-01T00:00:00Z
date_updated: 2021-01-12T06:57:05Z
day: '01'
doi: 10.1007/s00220-007-0307-2
extern: 1
intvolume: ' 275'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0610062
month: '10'
oa: 1
page: 479 - 489
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4555'
quality_controlled: 0
status: public
title: Stability of relativistic matter with magnetic fields for nuclear charges up
to the critical value
type: journal_article
volume: 275
year: '2007'
...
---
_id: '2372'
abstract:
- lang: eng
text: The increasing interest in the Müller density-matrix-functional theory has
led us to a systematic mathematical investigation of its properties. This functional
is similar to the Hartree-Fock (HF) functional, but with a modified exchange term
in which the square of the density matrix γ(x, x′) is replaced by the square of
γ1 2 (x, x′). After an extensive introductory discussion of density-matrix-functional
theory we show, among other things, that this functional is convex (unlike the
HF functional) and that energy minimizing γ 's have unique densities ρ(r), which
is a physically desirable property often absent in HF theory. We show that minimizers
exist if N≤Z, and derive various properties of the minimal energy and the corresponding
minimizers. We also give a precise statement about the equation for the orbitals
of γ, which is more complex than for HF theory. We state some open mathematical
questions about the theory together with conjectured solutions.
author:
- first_name: Rupert
full_name: Frank, Rupert L
last_name: Frank
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Heinz
full_name: Siedentop, Heinz K
last_name: Siedentop
citation:
ama: Frank R, Lieb É, Seiringer R, Siedentop H. Müller’s exchange-correlation energy
in density-matrix-functional theory. *Physical Review A - Atomic, Molecular,
and Optical Physics*. 2007;76(5). doi:10.1103/PhysRevA.76.052517
apa: Frank, R., Lieb, É., Seiringer, R., & Siedentop, H. (2007). Müller’s exchange-correlation
energy in density-matrix-functional theory. *Physical Review A - Atomic, Molecular,
and Optical Physics*. American Physical Society. https://doi.org/10.1103/PhysRevA.76.052517
chicago: Frank, Rupert, Élliott Lieb, Robert Seiringer, and Heinz Siedentop. “Müller’s
Exchange-Correlation Energy in Density-Matrix-Functional Theory.” *Physical
Review A - Atomic, Molecular, and Optical Physics*. American Physical Society,
2007. https://doi.org/10.1103/PhysRevA.76.052517.
ieee: R. Frank, É. Lieb, R. Seiringer, and H. Siedentop, “Müller’s exchange-correlation
energy in density-matrix-functional theory,” *Physical Review A - Atomic, Molecular,
and Optical Physics*, vol. 76, no. 5. American Physical Society, 2007.
ista: Frank R, Lieb É, Seiringer R, Siedentop H. 2007. Müller’s exchange-correlation
energy in density-matrix-functional theory. Physical Review A - Atomic, Molecular,
and Optical Physics. 76(5).
mla: Frank, Rupert, et al. “Müller’s Exchange-Correlation Energy in Density-Matrix-Functional
Theory.” *Physical Review A - Atomic, Molecular, and Optical Physics*, vol.
76, no. 5, American Physical Society, 2007, doi:10.1103/PhysRevA.76.052517.
short: R. Frank, É. Lieb, R. Seiringer, H. Siedentop, Physical Review A - Atomic,
Molecular, and Optical Physics 76 (2007).
date_created: 2018-12-11T11:57:17Z
date_published: 2007-11-30T00:00:00Z
date_updated: 2021-01-12T06:57:05Z
day: '30'
doi: 10.1103/PhysRevA.76.052517
extern: 1
intvolume: ' 76'
issue: '5'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0705.1587
month: '11'
oa: 1
publication: Physical Review A - Atomic, Molecular, and Optical Physics
publication_status: published
publisher: American Physical Society
publist_id: '4554'
quality_controlled: 0
status: public
title: Müller's exchange-correlation energy in density-matrix-functional theory
type: journal_article
volume: 76
year: '2007'
...
---
_id: '2373'
abstract:
- lang: eng
text: For the BCS equation with local two-body interaction λV(x), we give a rigorous
analysis of the asymptotic behavior of the critical temperature as γ"0. We
derive necessary and sufficient conditions onV(x) for the existence of a nontrivial
solution for all values of γ>0.
author:
- first_name: Rupert
full_name: Frank, Rupert L
last_name: Frank
- first_name: Christian
full_name: Hainzl, Christian
last_name: Hainzl
- first_name: Serguei
full_name: Naboko, Serguei N
last_name: Naboko
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Frank R, Hainzl C, Naboko S, Seiringer R. The critical temperature for the
BCS equation at weak coupling. *Journal of Geometric Analysis*. 2007;17(4):559-567.
doi:10.1007/BF02937429
apa: Frank, R., Hainzl, C., Naboko, S., & Seiringer, R. (2007). The critical
temperature for the BCS equation at weak coupling. *Journal of Geometric Analysis*.
Springer. https://doi.org/10.1007/BF02937429
chicago: Frank, Rupert, Christian Hainzl, Serguei Naboko, and Robert Seiringer.
“The Critical Temperature for the BCS Equation at Weak Coupling.” *Journal of
Geometric Analysis*. Springer, 2007. https://doi.org/10.1007/BF02937429.
ieee: R. Frank, C. Hainzl, S. Naboko, and R. Seiringer, “The critical temperature
for the BCS equation at weak coupling,” *Journal of Geometric Analysis*,
vol. 17, no. 4. Springer, pp. 559–567, 2007.
ista: Frank R, Hainzl C, Naboko S, Seiringer R. 2007. The critical temperature for
the BCS equation at weak coupling. Journal of Geometric Analysis. 17(4), 559–567.
mla: Frank, Rupert, et al. “The Critical Temperature for the BCS Equation at Weak
Coupling.” *Journal of Geometric Analysis*, vol. 17, no. 4, Springer, 2007,
pp. 559–67, doi:10.1007/BF02937429.
short: R. Frank, C. Hainzl, S. Naboko, R. Seiringer, Journal of Geometric Analysis
17 (2007) 559–567.
date_created: 2018-12-11T11:57:17Z
date_published: 2007-01-01T00:00:00Z
date_updated: 2021-01-12T06:57:05Z
day: '01'
doi: 10.1007/BF02937429
extern: 1
intvolume: ' 17'
issue: '4'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0704.3564
month: '01'
oa: 1
page: 559 - 567
publication: Journal of Geometric Analysis
publication_status: published
publisher: Springer
publist_id: '4553'
quality_controlled: 0
status: public
title: The critical temperature for the BCS equation at weak coupling
type: journal_article
volume: 17
year: '2007'
...
---
_id: '2375'
abstract:
- lang: eng
text: We give a Cwikel-Lieb-Rozenblum type bound on the number of bound states of
Schrödinger operators with matrix-valued potentials using the functional integral
method of Lieb. This significantly improves the constant in this inequality obtained
earlier by Hundertmark.
author:
- first_name: Rupert
full_name: Frank, Rupert L
last_name: Frank
- first_name: Élliott
full_name: Lieb, Élliott H
last_name: Lieb
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Frank R, Lieb É, Seiringer R. Number of bound states of Schrödinger operators
with matrix-valued potentials. *Letters in Mathematical Physics*. 2007;82(2-3):107-116.
doi:10.1007/s11005-007-0211-x
apa: Frank, R., Lieb, É., & Seiringer, R. (2007). Number of bound states of
Schrödinger operators with matrix-valued potentials. *Letters in Mathematical
Physics*. Springer. https://doi.org/10.1007/s11005-007-0211-x
chicago: Frank, Rupert, Élliott Lieb, and Robert Seiringer. “Number of Bound States
of Schrödinger Operators with Matrix-Valued Potentials.” *Letters in Mathematical
Physics*. Springer, 2007. https://doi.org/10.1007/s11005-007-0211-x.
ieee: R. Frank, É. Lieb, and R. Seiringer, “Number of bound states of Schrödinger
operators with matrix-valued potentials,” *Letters in Mathematical Physics*,
vol. 82, no. 2–3. Springer, pp. 107–116, 2007.
ista: Frank R, Lieb É, Seiringer R. 2007. Number of bound states of Schrödinger
operators with matrix-valued potentials. Letters in Mathematical Physics. 82(2–3),
107–116.
mla: Frank, Rupert, et al. “Number of Bound States of Schrödinger Operators with
Matrix-Valued Potentials.” *Letters in Mathematical Physics*, vol. 82, no.
2–3, Springer, 2007, pp. 107–16, doi:10.1007/s11005-007-0211-x.
short: R. Frank, É. Lieb, R. Seiringer, Letters in Mathematical Physics 82 (2007)
107–116.
date_created: 2018-12-11T11:57:18Z
date_published: 2007-12-01T00:00:00Z
date_updated: 2021-01-12T06:57:06Z
day: '01'
doi: 10.1007/s11005-007-0211-x
extern: 1
intvolume: ' 82'
issue: 2-3
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0710.1877
month: '12'
oa: 1
page: 107 - 116
publication: Letters in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4552'
quality_controlled: 0
status: public
title: Number of bound states of Schrödinger operators with matrix-valued potentials
type: journal_article
volume: 82
year: '2007'
...