---
_id: '2340'
abstract:
- lang: eng
text: |2
Recent experimental breakthroughs in the treatment of dilute Bose gases have renewed interest in their quantum mechanical description, respectively in approximations to it. The ground state properties of dilute Bose gases confined in external potentials and interacting via repulsive short range forces are usually described by means of the Gross-Pitaevskii energy functional. In joint work with Elliott H. Lieb and Jakob Yngvason its status as an approximation for the quantum mechanical many-body ground state problem has recently been rigorously clarified. We present a summary of this work, for both the two-and three-dimensional case.
alternative_title:
- 'Operator Theory: Advances and Applications'
author:
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: 'Seiringer R. Bosons in a trap: Asymptotic exactness of the Gross-Pitaevskii
ground state energy formula. In: Demuth M, Schultze B, eds. Vol 126. Birkhäuser;
2001:307-314. doi:10.1007/978-3-0348-8231-6'
apa: 'Seiringer, R. (2001). Bosons in a trap: Asymptotic exactness of the Gross-Pitaevskii
ground state energy formula. In M. Demuth & B. Schultze (Eds.) (Vol. 126,
pp. 307–314). Presented at the PDE: Partial Differential Equations and Spectral
Theory, Birkhäuser. https://doi.org/10.1007/978-3-0348-8231-6'
chicago: 'Seiringer, Robert. “Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii
Ground State Energy Formula.” edited by Michael Demuth and Bert Schultze, 126:307–14.
Birkhäuser, 2001. https://doi.org/10.1007/978-3-0348-8231-6.'
ieee: 'R. Seiringer, “Bosons in a trap: Asymptotic exactness of the Gross-Pitaevskii
ground state energy formula,” presented at the PDE: Partial Differential Equations
and Spectral Theory, 2001, vol. 126, pp. 307–314.'
ista: 'Seiringer R. 2001. Bosons in a trap: Asymptotic exactness of the Gross-Pitaevskii
ground state energy formula. PDE: Partial Differential Equations and Spectral
Theory, Operator Theory: Advances and Applications, vol. 126. 307–314.'
mla: 'Seiringer, Robert. *Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii
Ground State Energy Formula*. Edited by Michael Demuth and Bert Schultze, vol.
126, Birkhäuser, 2001, pp. 307–14, doi:10.1007/978-3-0348-8231-6.'
short: R. Seiringer, in:, M. Demuth, B. Schultze (Eds.), Birkhäuser, 2001, pp. 307–314.
conference:
name: 'PDE: Partial Differential Equations and Spectral Theory'
date_created: 2018-12-11T11:57:05Z
date_published: 2001-01-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1007/978-3-0348-8231-6
editor:
- first_name: Michael
full_name: Demuth, Michael
last_name: Demuth
- first_name: Bert
full_name: Schultze, Bert-Wolfgang
last_name: Schultze
extern: 1
intvolume: ' 126'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0010006
month: '01'
oa: 1
page: 307 - 314
publication_status: published
publisher: Birkhäuser
publist_id: '4586'
quality_controlled: 0
status: public
title: 'Bosons in a trap: Asymptotic exactness of the Gross-Pitaevskii ground state
energy formula'
type: conference
volume: 126
year: '2001'
...
---
_id: '2341'
abstract:
- lang: eng
text: We study the ground state properties of an atom with nuclear charge Z and
N bosonic "electrons" in the presence of a homogeneous magnetic field
B. We investigate the mean field limit N→∞ with N / Z fixed, and identify three
different asymptotic regions, according to B≪Z2,B∼Z2,andB≫Z2 . In Region 1 standard
Hartree theory is applicable. Region 3 is described by a one-dimensional functional,
which is identical to the so-called Hyper-Strong functional introduced by Lieb,
Solovej and Yngvason for atoms with fermionic electrons in the region B≫Z3 ; i.e.,
for very strong magnetic fields the ground state properties of atoms are independent
of statistics. For Region 2 we introduce a general magnetic Hartree functional,
which is studied in detail. It is shown that in the special case of an atom it
can be restricted to the subspace of zero angular momentum parallel to the magnetic
field, which simplifies the theory considerably. The functional reproduces the
energy and the one-particle reduced density matrix for the full N-particle ground
state to leading order in N, and it implies the description of the other regions
as limiting cases.
author:
- first_name: Bernhard
full_name: Baumgartner, Bernhard
last_name: Baumgartner
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Baumgartner B, Seiringer R. Atoms with bosonic "electrons"
in strong magnetic fields. *Annales Henri Poincare*. 2001;2(1):41-76. doi:10.1007/PL00001032
apa: Baumgartner, B., & Seiringer, R. (2001). Atoms with bosonic "electrons"
in strong magnetic fields. *Annales Henri Poincare*, *2*(1), 41–76.
https://doi.org/10.1007/PL00001032
chicago: 'Baumgartner, Bernhard, and Robert Seiringer. “Atoms with Bosonic "Electrons"
in Strong Magnetic Fields.” *Annales Henri Poincare* 2, no. 1 (2001): 41–76.
https://doi.org/10.1007/PL00001032.'
ieee: B. Baumgartner and R. Seiringer, “Atoms with bosonic "electrons"
in strong magnetic fields,” *Annales Henri Poincare*, vol. 2, no. 1, pp.
41–76, 2001.
ista: Baumgartner B, Seiringer R. 2001. Atoms with bosonic "electrons"
in strong magnetic fields. Annales Henri Poincare. 2(1), 41–76.
mla: Baumgartner, Bernhard, and Robert Seiringer. “Atoms with Bosonic "Electrons"
in Strong Magnetic Fields.” *Annales Henri Poincare*, vol. 2, no. 1, Birkhäuser,
2001, pp. 41–76, doi:10.1007/PL00001032.
short: B. Baumgartner, R. Seiringer, Annales Henri Poincare 2 (2001) 41–76.
date_created: 2018-12-11T11:57:06Z
date_published: 2001-02-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1007/PL00001032
extern: 1
intvolume: ' 2'
issue: '1'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0007007
month: '02'
oa: 1
page: 41 - 76
publication: Annales Henri Poincare
publication_status: published
publisher: Birkhäuser
publist_id: '4585'
quality_controlled: 0
status: public
title: Atoms with bosonic "electrons" in strong magnetic fields
type: journal_article
volume: 2
year: '2001'
...
---
_id: '2345'
abstract:
- lang: eng
text: We give upper bounds for the number of spin-1/2 particles that can be bound
to a nucleus of charge Z in the presence of a magnetic field B, including the
spin-field coupling. We use Lieb's strategy, which is known to yield Nc < 2Z
+ 1 for magnetic fields that go to zero at infinity, ignoring the spin-field interaction.
For particles with fermionic statistics in a homogeneous magnetic field our upper
bound has an additional term of the order of Z × min {(B/Z3)2/5, 1 + | 1n(B/Z3)|2}.
author:
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: 'Seiringer R. On the maximal ionization of atoms in strong magnetic fields.
*Journal of Physics A: Mathematical and General*. 2001;34(9):1943-1948. doi:10.1088/0305-4470/34/9/311'
apa: 'Seiringer, R. (2001). On the maximal ionization of atoms in strong magnetic
fields. *Journal of Physics A: Mathematical and General*, *34*(9), 1943–1948.
https://doi.org/10.1088/0305-4470/34/9/311'
chicago: 'Seiringer, Robert. “On the Maximal Ionization of Atoms in Strong Magnetic
Fields.” *Journal of Physics A: Mathematical and General* 34, no. 9 (2001):
1943–48. https://doi.org/10.1088/0305-4470/34/9/311.'
ieee: 'R. Seiringer, “On the maximal ionization of atoms in strong magnetic fields,”
*Journal of Physics A: Mathematical and General*, vol. 34, no. 9, pp. 1943–1948,
2001.'
ista: 'Seiringer R. 2001. On the maximal ionization of atoms in strong magnetic
fields. Journal of Physics A: Mathematical and General. 34(9), 1943–1948.'
mla: 'Seiringer, Robert. “On the Maximal Ionization of Atoms in Strong Magnetic
Fields.” *Journal of Physics A: Mathematical and General*, vol. 34, no. 9,
IOP Publishing Ltd., 2001, pp. 1943–48, doi:10.1088/0305-4470/34/9/311.'
short: 'R. Seiringer, Journal of Physics A: Mathematical and General 34 (2001) 1943–1948.'
date_created: 2018-12-11T11:57:07Z
date_published: 2001-03-09T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '09'
doi: 10.1088/0305-4470/34/9/311
extern: 1
intvolume: ' 34'
issue: '9'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0006002
month: '03'
oa: 1
page: 1943 - 1948
publication: 'Journal of Physics A: Mathematical and General'
publication_status: published
publisher: IOP Publishing Ltd.
publist_id: '4580'
quality_controlled: 0
status: public
title: On the maximal ionization of atoms in strong magnetic fields
type: journal_article
volume: 34
year: '2001'
...
---
_id: '2346'
abstract:
- lang: eng
text: By means of a generalization of the Fefferman - de la Llave decomposition
we derive a general lower bound on the interaction energy of one-dimensional quantum
systems. We apply this result to a specific class of lowest Landau band wave functions.
author:
- first_name: Christian
full_name: Hainzl, Christian
last_name: Hainzl
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Hainzl C, Seiringer R. Bounds on one-dimensional exchange energies with application
to lowest Landau band quantum mechanics. *Letters in Mathematical Physics*.
2001;55(2):133-142. doi:10.1023/A:1010951905548
apa: Hainzl, C., & Seiringer, R. (2001). Bounds on one-dimensional exchange
energies with application to lowest Landau band quantum mechanics. *Letters
in Mathematical Physics*, *55*(2), 133–142. https://doi.org/10.1023/A:1010951905548
chicago: 'Hainzl, Christian, and Robert Seiringer. “Bounds on One-Dimensional Exchange
Energies with Application to Lowest Landau Band Quantum Mechanics.” *Letters
in Mathematical Physics* 55, no. 2 (2001): 133–42. https://doi.org/10.1023/A:1010951905548.'
ieee: C. Hainzl and R. Seiringer, “Bounds on one-dimensional exchange energies with
application to lowest Landau band quantum mechanics,” *Letters in Mathematical
Physics*, vol. 55, no. 2, pp. 133–142, 2001.
ista: Hainzl C, Seiringer R. 2001. Bounds on one-dimensional exchange energies with
application to lowest Landau band quantum mechanics. Letters in Mathematical Physics.
55(2), 133–142.
mla: Hainzl, Christian, and Robert Seiringer. “Bounds on One-Dimensional Exchange
Energies with Application to Lowest Landau Band Quantum Mechanics.” *Letters
in Mathematical Physics*, vol. 55, no. 2, Springer, 2001, pp. 133–42, doi:10.1023/A:1010951905548.
short: C. Hainzl, R. Seiringer, Letters in Mathematical Physics 55 (2001) 133–142.
date_created: 2018-12-11T11:57:07Z
date_published: 2001-02-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1023/A:1010951905548
extern: 1
intvolume: ' 55'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/cond-mat/0102118
month: '02'
oa: 1
page: 133 - 142
publication: Letters in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4581'
quality_controlled: 0
status: public
title: Bounds on one-dimensional exchange energies with application to lowest Landau
band quantum mechanics
type: journal_article
volume: 55
year: '2001'
...
---
_id: '2347'
abstract:
- lang: eng
text: We consider the ground state properties of an inhomogeneous two-dimensional
Bose gas with a repulsive, short range pair interaction and an external confining
potential. In the limit when the particle number N is large but ρ̄a2 is small,
where ρ̄ is the average particle density and a the scattering length, the ground
state energy and density are rigorously shown to be given to leading order by
a Gross-Pitaevskii (GP) energy functional with a coupling constant g ∼ 1/| 1n(ρ̄a2)|.
In contrast to the 3D case the coupling constant depends on N through the mean
density. The GP energy per particle depends only on Ng. In 2D this parameter is
typically so large that the gradient term in the GP energy functional is negligible
and the simpler description by a Thomas-Fermi type functional is adequate.
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. A rigorous derivation of the Gross-Pitaevskii
energy functional for a two-dimensional Bose gas. *Communications in Mathematical
Physics*. 2001;224(1):17-31. doi:10.1007/s002200100533
apa: Lieb, É., Seiringer, R., & Yngvason, J. (2001). A rigorous derivation of
the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. *Communications
in Mathematical Physics*, *224*(1), 17–31. https://doi.org/10.1007/s002200100533
chicago: 'Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “A Rigorous Derivation
of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas.” *Communications
in Mathematical Physics* 224, no. 1 (2001): 17–31. https://doi.org/10.1007/s002200100533.'
ieee: É. Lieb, R. Seiringer, and J. Yngvason, “A rigorous derivation of the Gross-Pitaevskii
energy functional for a two-dimensional Bose gas,” *Communications in Mathematical
Physics*, vol. 224, no. 1, pp. 17–31, 2001.
ista: Lieb É, Seiringer R, Yngvason J. 2001. A rigorous derivation of the Gross-Pitaevskii
energy functional for a two-dimensional Bose gas. Communications in Mathematical
Physics. 224(1), 17–31.
mla: Lieb, Élliott, et al. “A Rigorous Derivation of the Gross-Pitaevskii Energy
Functional for a Two-Dimensional Bose Gas.” *Communications in Mathematical
Physics*, vol. 224, no. 1, Springer, 2001, pp. 17–31, doi:10.1007/s002200100533.
short: É. Lieb, R. Seiringer, J. Yngvason, Communications in Mathematical Physics
224 (2001) 17–31.
date_created: 2018-12-11T11:57:08Z
date_published: 2001-11-01T00:00:00Z
date_updated: 2020-07-14T12:45:39Z
day: '01'
doi: 10.1007/s002200100533
extern: 1
intvolume: ' 224'
issue: '1'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/cond-mat/0005026
month: '11'
oa: 1
page: 17 - 31
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4579'
quality_controlled: 0
status: public
title: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional
Bose gas
type: journal_article
volume: 224
year: '2001'
...