---
_id: '2331'
abstract:
- lang: eng
text: We present a review of recent work on the mathematical aspects of the BCS
gap equation, covering our results of Ref. 9 as well our recent joint work with
Hamza and Solovej and with Frank and Naboko, respectively. In addition, we mention
some related new results.
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. Spectral properties of the BCS gap equation of superfluidity.
In: World Scientific Publishing; 2008:117-136. doi:10.1142/9789812832382_0009'
apa: 'Hainzl, C., & Seiringer, R. (2008). Spectral properties of the BCS gap
equation of superfluidity (pp. 117–136). Presented at the QMath: Mathematical
Results in Quantum Physics, World Scientific Publishing. https://doi.org/10.1142/9789812832382_0009'
chicago: Hainzl, Christian, and Robert Seiringer. “ Spectral Properties of the BCS
Gap Equation of Superfluidity,” 117–36. World Scientific Publishing, 2008. https://doi.org/10.1142/9789812832382_0009.
ieee: 'C. Hainzl and R. Seiringer, “ Spectral properties of the BCS gap equation
of superfluidity,” presented at the QMath: Mathematical Results in Quantum Physics,
2008, pp. 117–136.'
ista: 'Hainzl C, Seiringer R. 2008. Spectral properties of the BCS gap equation
of superfluidity. QMath: Mathematical Results in Quantum Physics, 117–136.'
mla: Hainzl, Christian, and Robert Seiringer. * Spectral Properties of the BCS
Gap Equation of Superfluidity*. World Scientific Publishing, 2008, pp. 117–36,
doi:10.1142/9789812832382_0009.
short: C. Hainzl, R. Seiringer, in:, World Scientific Publishing, 2008, pp. 117–136.
conference:
name: 'QMath: Mathematical Results in Quantum Physics'
date_created: 2018-12-11T11:57:02Z
date_published: 2008-08-01T00:00:00Z
date_updated: 2021-01-12T06:56:50Z
day: '01'
doi: 10.1142/9789812832382_0009
extern: 1
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0802.0446
month: '08'
oa: 1
page: 117 - 136
publication_status: published
publisher: World Scientific Publishing
publist_id: '4595'
quality_controlled: 0
status: public
title: ' Spectral properties of the BCS gap equation of superfluidity'
type: conference
year: '2008'
...
---
_id: '2332'
abstract:
- lang: eng
text: We present a rigorous proof of the appearance of quantized vortices in dilute
trapped Bose gases with repulsive two-body interactions subject to rotation, which
was obtained recently in joint work with Elliott Lieb.14 Starting from the many-body
Schrödinger equation, we show that the ground state of such gases is, in a suitable
limit, well described by the nonlinear Gross-Pitaevskii equation. In the case
of axially symmetric traps, our results show that the appearance of quantized
vortices causes spontaneous symmetry breaking in the ground state.
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. Vortices and Spontaneous Symmetry Breaking in Rotating Bose Gases.
In: World Scientific Publishing; 2008:241-254. doi:10.1142/9789812832382_0017'
apa: 'Seiringer, R. (2008). Vortices and Spontaneous Symmetry Breaking in Rotating
Bose Gases (pp. 241–254). Presented at the QMath: Mathematical Results in Quantum
Physics, World Scientific Publishing. https://doi.org/10.1142/9789812832382_0017'
chicago: Seiringer, Robert. “Vortices and Spontaneous Symmetry Breaking in Rotating
Bose Gases,” 241–54. World Scientific Publishing, 2008. https://doi.org/10.1142/9789812832382_0017.
ieee: 'R. Seiringer, “Vortices and Spontaneous Symmetry Breaking in Rotating Bose
Gases,” presented at the QMath: Mathematical Results in Quantum Physics, 2008,
pp. 241–254.'
ista: 'Seiringer R. 2008. Vortices and Spontaneous Symmetry Breaking in Rotating
Bose Gases. QMath: Mathematical Results in Quantum Physics, 241–254.'
mla: Seiringer, Robert. *Vortices and Spontaneous Symmetry Breaking in Rotating
Bose Gases*. World Scientific Publishing, 2008, pp. 241–54, doi:10.1142/9789812832382_0017.
short: R. Seiringer, in:, World Scientific Publishing, 2008, pp. 241–254.
conference:
name: 'QMath: Mathematical Results in Quantum Physics'
date_created: 2018-12-11T11:57:02Z
date_published: 2008-12-30T00:00:00Z
date_updated: 2021-01-12T06:56:50Z
day: '30'
doi: 10.1142/9789812832382_0017
extern: 1
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0801.0427
month: '12'
oa: 1
page: 241 - 254
publication_status: published
publisher: World Scientific Publishing
publist_id: '4594'
quality_controlled: 0
status: public
title: Vortices and Spontaneous Symmetry Breaking in Rotating Bose Gases
type: conference
year: '2008'
...
---
_id: '2374'
abstract:
- lang: eng
text: A lower bound is derived on the free energy (per unit volume) of a homogeneous
Bose gas at density Q and temperature T. In the dilute regime, i.e., when a3 1,
where a denotes the scattering length of the pair-interaction potential, our bound
differs to leading order from the expression for non-interacting particles by
the term 4πa(2 2}-[ - c]2+). Here, c(T) denotes the critical density for Bose-Einstein
condensation (for the non-interacting gas), and [ · ]+ = max{ ·, 0} denotes the
positive part. Our bound is uniform in the temperature up to temperatures of the
order of the critical temperature, i.e., T ~ 2/3 or smaller. One of the key ingredients
in the proof is the use of coherent states to extend the method introduced in
[17] for estimating correlations to temperatures below the critical one.
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. Free energy of a dilute Bose gas: Lower bound. *Communications
in Mathematical Physics*. 2008;279(3):595-636. doi:10.1007/s00220-008-0428-2'
apa: 'Seiringer, R. (2008). Free energy of a dilute Bose gas: Lower bound. *Communications
in Mathematical Physics*. Springer. https://doi.org/10.1007/s00220-008-0428-2'
chicago: 'Seiringer, Robert. “Free Energy of a Dilute Bose Gas: Lower Bound.” *Communications
in Mathematical Physics*. Springer, 2008. https://doi.org/10.1007/s00220-008-0428-2.'
ieee: 'R. Seiringer, “Free energy of a dilute Bose gas: Lower bound,” *Communications
in Mathematical Physics*, vol. 279, no. 3. Springer, pp. 595–636, 2008.'
ista: 'Seiringer R. 2008. Free energy of a dilute Bose gas: Lower bound. Communications
in Mathematical Physics. 279(3), 595–636.'
mla: 'Seiringer, Robert. “Free Energy of a Dilute Bose Gas: Lower Bound.” *Communications
in Mathematical Physics*, vol. 279, no. 3, Springer, 2008, pp. 595–636, doi:10.1007/s00220-008-0428-2.'
short: R. Seiringer, Communications in Mathematical Physics 279 (2008) 595–636.
date_created: 2018-12-11T11:57:17Z
date_published: 2008-05-01T00:00:00Z
date_updated: 2021-01-12T06:57:06Z
day: '01'
doi: 10.1007/s00220-008-0428-2
extern: 1
intvolume: ' 279'
issue: '3'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0608069
month: '05'
oa: 1
page: 595 - 636
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4551'
quality_controlled: 0
status: public
title: 'Free energy of a dilute Bose gas: Lower bound'
type: journal_article
volume: 279
year: '2008'
...
---
_id: '2376'
abstract:
- lang: eng
text: We derive upper and lower bounds on the critical temperature Tc and the energy
gap Ξ (at zero temperature) for the BCS gap equation, describing spin- 1 2 fermions
interacting via a local two-body interaction potential λV(x). At weak coupling
λ 1 and under appropriate assumptions on V(x), our bounds show that Tc ∼A exp(-B/λ)
and Ξ∼C exp(-B/λ) for some explicit coefficients A, B, and C depending on the
interaction V(x) and the chemical potential μ. The ratio A/C turns out to be a
universal constant, independent of both V(x) and μ. Our analysis is valid for
any μ; for small μ, or low density, our formulas reduce to well-known expressions
involving the scattering length of V(x).
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. Critical temperature and energy gap for the BCS equation.
*Physical Review B - Condensed Matter and Materials Physics*. 2008;77(18).
doi:10.1103/PhysRevB.77.184517
apa: Hainzl, C., & Seiringer, R. (2008). Critical temperature and energy gap
for the BCS equation. *Physical Review B - Condensed Matter and Materials Physics*.
American Physical Society. https://doi.org/10.1103/PhysRevB.77.184517
chicago: Hainzl, Christian, and Robert Seiringer. “Critical Temperature and Energy
Gap for the BCS Equation.” *Physical Review B - Condensed Matter and Materials
Physics*. American Physical Society, 2008. https://doi.org/10.1103/PhysRevB.77.184517.
ieee: C. Hainzl and R. Seiringer, “Critical temperature and energy gap for the BCS
equation,” *Physical Review B - Condensed Matter and Materials Physics*,
vol. 77, no. 18. American Physical Society, 2008.
ista: Hainzl C, Seiringer R. 2008. Critical temperature and energy gap for the BCS
equation. Physical Review B - Condensed Matter and Materials Physics. 77(18).
mla: Hainzl, Christian, and Robert Seiringer. “Critical Temperature and Energy Gap
for the BCS Equation.” *Physical Review B - Condensed Matter and Materials Physics*,
vol. 77, no. 18, American Physical Society, 2008, doi:10.1103/PhysRevB.77.184517.
short: C. Hainzl, R. Seiringer, Physical Review B - Condensed Matter and Materials
Physics 77 (2008).
date_created: 2018-12-11T11:57:18Z
date_published: 2008-05-28T00:00:00Z
date_updated: 2021-01-12T06:57:06Z
day: '28'
doi: 10.1103/PhysRevB.77.184517
extern: 1
intvolume: ' 77'
issue: '18'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0801.4159
month: '05'
oa: 1
publication: Physical Review B - Condensed Matter and Materials Physics
publication_status: published
publisher: American Physical Society
publist_id: '4550'
quality_controlled: 0
status: public
title: Critical temperature and energy gap for the BCS equation
type: journal_article
volume: 77
year: '2008'
...
---
_id: '2377'
abstract:
- lang: eng
text: We prove that the critical temperature for the BCS gap equation is given by
T c = μ ( 8\π e γ-2+ o(1)) e π/(2μa) in the low density limit μ→ 0, with γ denoting
Euler's constant. The formula holds for a suitable class of interaction potentials
with negative scattering length a in the absence of bound states.
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. The BCS critical temperature for potentials with negative
scattering length. *Letters in Mathematical Physics*. 2008;84(2-3):99-107.
doi:10.1007/s11005-008-0242-y
apa: Hainzl, C., & Seiringer, R. (2008). The BCS critical temperature for potentials
with negative scattering length. *Letters in Mathematical Physics*. Springer.
https://doi.org/10.1007/s11005-008-0242-y
chicago: Hainzl, Christian, and Robert Seiringer. “The BCS Critical Temperature
for Potentials with Negative Scattering Length.” *Letters in Mathematical Physics*.
Springer, 2008. https://doi.org/10.1007/s11005-008-0242-y.
ieee: C. Hainzl and R. Seiringer, “The BCS critical temperature for potentials with
negative scattering length,” *Letters in Mathematical Physics*, vol. 84,
no. 2–3. Springer, pp. 99–107, 2008.
ista: Hainzl C, Seiringer R. 2008. The BCS critical temperature for potentials with
negative scattering length. Letters in Mathematical Physics. 84(2–3), 99–107.
mla: Hainzl, Christian, and Robert Seiringer. “The BCS Critical Temperature for
Potentials with Negative Scattering Length.” *Letters in Mathematical Physics*,
vol. 84, no. 2–3, Springer, 2008, pp. 99–107, doi:10.1007/s11005-008-0242-y.
short: C. Hainzl, R. Seiringer, Letters in Mathematical Physics 84 (2008) 99–107.
date_created: 2018-12-11T11:57:19Z
date_published: 2008-06-01T00:00:00Z
date_updated: 2021-01-12T06:57:07Z
day: '01'
doi: 10.1007/s11005-008-0242-y
extern: 1
intvolume: ' 84'
issue: 2-3
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0803.3324
month: '06'
oa: 1
page: 99 - 107
publication: Letters in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4548'
quality_controlled: 0
status: public
title: The BCS critical temperature for potentials with negative scattering length
type: journal_article
volume: 84
year: '2008'
...
---
_id: '2378'
abstract:
- lang: eng
text: We derive a lower bound on the ground state energy of the Hubbard model for
given value of the total spin. In combination with the upper bound derived previously
by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the
low density limit the leading order correction compared to the ground state energy
of a non-interacting lattice Fermi gas is given by 8πaσ uσ d , where σ u(d) denotes
the density of the spin-up (down) particles, and a is the scattering length of
the contact interaction potential. This result extends previous work on the corresponding
continuum model to the lattice case.
author:
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Seiringer R, Yin J. Ground state energy of the low density hubbard model. *Journal
of Statistical Physics*. 2008;131(6):1139-1154. doi:10.1007/s10955-008-9527-x
apa: Seiringer, R., & Yin, J. (2008). Ground state energy of the low density
hubbard model. *Journal of Statistical Physics*. Springer. https://doi.org/10.1007/s10955-008-9527-x
chicago: Seiringer, Robert, and Jun Yin. “Ground State Energy of the Low Density
Hubbard Model.” *Journal of Statistical Physics*. Springer, 2008. https://doi.org/10.1007/s10955-008-9527-x.
ieee: R. Seiringer and J. Yin, “Ground state energy of the low density hubbard model,”
*Journal of Statistical Physics*, vol. 131, no. 6. Springer, pp. 1139–1154,
2008.
ista: Seiringer R, Yin J. 2008. Ground state energy of the low density hubbard model.
Journal of Statistical Physics. 131(6), 1139–1154.
mla: Seiringer, Robert, and Jun Yin. “Ground State Energy of the Low Density Hubbard
Model.” *Journal of Statistical Physics*, vol. 131, no. 6, Springer, 2008,
pp. 1139–54, doi:10.1007/s10955-008-9527-x.
short: R. Seiringer, J. Yin, Journal of Statistical Physics 131 (2008) 1139–1154.
date_created: 2018-12-11T11:57:19Z
date_published: 2008-06-01T00:00:00Z
date_updated: 2021-01-12T06:57:07Z
day: '01'
doi: 10.1007/s10955-008-9527-x
extern: 1
intvolume: ' 131'
issue: '6'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0712.2810
month: '06'
oa: 1
page: 1139 - 1154
publication: Journal of Statistical Physics
publication_status: published
publisher: Springer
publist_id: '4549'
quality_controlled: 0
status: public
title: Ground state energy of the low density hubbard model
type: journal_article
volume: 131
year: '2008'
...
---
_id: '2379'
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. Hardy-Lieb-Thirring inequalities for fractional
Schrödinger operators. *Journal of the American Mathematical Society*. 2008;21(4):925-950.
doi:10.1090/S0894-0347-07-00582-6
apa: Frank, R., Lieb, É., & Seiringer, R. (2008). Hardy-Lieb-Thirring inequalities
for fractional Schrödinger operators. *Journal of the American Mathematical
Society*. American Mathematical Society. https://doi.org/10.1090/S0894-0347-07-00582-6
chicago: Frank, Rupert, Élliott Lieb, and Robert Seiringer. “Hardy-Lieb-Thirring
Inequalities for Fractional Schrödinger Operators.” *Journal of the American
Mathematical Society*. American Mathematical Society, 2008. https://doi.org/10.1090/S0894-0347-07-00582-6.
ieee: R. Frank, É. Lieb, and R. Seiringer, “Hardy-Lieb-Thirring inequalities for
fractional Schrödinger operators,” *Journal of the American Mathematical Society*,
vol. 21, no. 4. American Mathematical Society, pp. 925–950, 2008.
ista: Frank R, Lieb É, Seiringer R. 2008. Hardy-Lieb-Thirring inequalities for fractional
Schrödinger operators. Journal of the American Mathematical Society. 21(4), 925–950.
mla: Frank, Rupert, et al. “Hardy-Lieb-Thirring Inequalities for Fractional Schrödinger
Operators.” *Journal of the American Mathematical Society*, vol. 21, no.
4, American Mathematical Society, 2008, pp. 925–50, doi:10.1090/S0894-0347-07-00582-6.
short: R. Frank, É. Lieb, R. Seiringer, Journal of the American Mathematical Society
21 (2008) 925–950.
date_created: 2018-12-11T11:57:19Z
date_published: 2008-01-01T00:00:00Z
date_updated: 2021-01-12T06:57:07Z
day: '01'
doi: 10.1090/S0894-0347-07-00582-6
extern: 1
intvolume: ' 21'
issue: '4'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math/0610593
month: '01'
oa: 1
page: 925 - 950
publication: Journal of the American Mathematical Society
publication_status: published
publisher: American Mathematical Society
publist_id: '4546'
quality_controlled: 0
status: public
title: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
type: journal_article
volume: 21
year: '2008'
...
---
_id: '2380'
abstract:
- lang: eng
text: The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed
attention as a description of fermionic gases interacting with local pairwise
interactions. We present here a rigorous analysis of the BCS functional for general
pair interaction potentials. For both zero and positive temperature, we show that
the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent
to the existence of a negative eigenvalue of a certain linear operator. From this
we conclude the existence of a critical temperature below which the BCS pairing
wave function does not vanish identically. For attractive potentials, we prove
that the critical temperature is non-zero and exponentially small in the strength
of the potential.
author:
- first_name: Christian
full_name: Hainzl, Christian
last_name: Hainzl
- first_name: Eman
full_name: Hamza, Eman
last_name: Hamza
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jan
full_name: Solovej, Jan P
last_name: Solovej
citation:
ama: Hainzl C, Hamza E, Seiringer R, Solovej J. The BCS functional for general pair
interactions. *Communications in Mathematical Physics*. 2008;281(2):349-367.
doi:10.1007/s00220-008-0489-2
apa: Hainzl, C., Hamza, E., Seiringer, R., & Solovej, J. (2008). The BCS functional
for general pair interactions. *Communications in Mathematical Physics*.
Springer. https://doi.org/10.1007/s00220-008-0489-2
chicago: Hainzl, Christian, Eman Hamza, Robert Seiringer, and Jan Solovej. “The
BCS Functional for General Pair Interactions.” *Communications in Mathematical
Physics*. Springer, 2008. https://doi.org/10.1007/s00220-008-0489-2.
ieee: C. Hainzl, E. Hamza, R. Seiringer, and J. Solovej, “The BCS functional for
general pair interactions,” *Communications in Mathematical Physics*, vol.
281, no. 2. Springer, pp. 349–367, 2008.
ista: Hainzl C, Hamza E, Seiringer R, Solovej J. 2008. The BCS functional for general
pair interactions. Communications in Mathematical Physics. 281(2), 349–367.
mla: Hainzl, Christian, et al. “The BCS Functional for General Pair Interactions.”
*Communications in Mathematical Physics*, vol. 281, no. 2, Springer, 2008,
pp. 349–67, doi:10.1007/s00220-008-0489-2.
short: C. Hainzl, E. Hamza, R. Seiringer, J. Solovej, Communications in Mathematical
Physics 281 (2008) 349–367.
date_created: 2018-12-11T11:57:20Z
date_published: 2008-07-01T00:00:00Z
date_updated: 2021-01-12T06:57:08Z
day: '01'
doi: 10.1007/s00220-008-0489-2
extern: 1
intvolume: ' 281'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/math-ph/0703086
month: '07'
oa: 1
page: 349 - 367
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4547'
quality_controlled: 0
status: public
title: The BCS functional for general pair interactions
type: journal_article
volume: 281
year: '2008'
...
---
_id: '2381'
abstract:
- lang: eng
text: We determine the sharp constant in the Hardy inequality for fractional Sobolev
spaces. To do so, we develop a non-linear and non-local version of the ground
state representation, which even yields a remainder term. From the sharp Hardy
inequality we deduce the sharp constant in a Sobolev embedding which is optimal
in the Lorentz scale. In the appendix, we characterize the cases of equality in
the rearrangement inequality in fractional Sobolev spaces.
author:
- first_name: Rupert
full_name: Frank, Rupert L
last_name: Frank
- 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, Seiringer R. Non-linear ground state representations and sharp Hardy
inequalities. *Journal of Functional Analysis*. 2008;255(12):3407-3430. doi:10.1016/j.jfa.2008.05.015
apa: Frank, R., & Seiringer, R. (2008). Non-linear ground state representations
and sharp Hardy inequalities. *Journal of Functional Analysis*. Academic
Press. https://doi.org/10.1016/j.jfa.2008.05.015
chicago: Frank, Rupert, and Robert Seiringer. “Non-Linear Ground State Representations
and Sharp Hardy Inequalities.” *Journal of Functional Analysis*. Academic
Press, 2008. https://doi.org/10.1016/j.jfa.2008.05.015.
ieee: R. Frank and R. Seiringer, “Non-linear ground state representations and sharp
Hardy inequalities,” *Journal of Functional Analysis*, vol. 255, no. 12.
Academic Press, pp. 3407–3430, 2008.
ista: Frank R, Seiringer R. 2008. Non-linear ground state representations and sharp
Hardy inequalities. Journal of Functional Analysis. 255(12), 3407–3430.
mla: Frank, Rupert, and Robert Seiringer. “Non-Linear Ground State Representations
and Sharp Hardy Inequalities.” *Journal of Functional Analysis*, vol. 255,
no. 12, Academic Press, 2008, pp. 3407–30, doi:10.1016/j.jfa.2008.05.015.
short: R. Frank, R. Seiringer, Journal of Functional Analysis 255 (2008) 3407–3430.
date_created: 2018-12-11T11:57:20Z
date_published: 2008-12-15T00:00:00Z
date_updated: 2021-01-12T06:57:08Z
day: '15'
doi: 10.1016/j.jfa.2008.05.015
extern: 1
intvolume: ' 255'
issue: '12'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0803.0503
month: '12'
oa: 1
page: 3407 - 3430
publication: Journal of Functional Analysis
publication_status: published
publisher: Academic Press
publist_id: '4543'
quality_controlled: 0
status: public
title: Non-linear ground state representations and sharp Hardy inequalities
type: journal_article
volume: 255
year: '2008'
...
---
_id: '2382'
abstract:
- lang: eng
text: We show that the Lieb-Liniger model for one-dimensional bosons with repulsive
δ-function interaction can be rigorously derived via a scaling limit from a dilute
three-dimensional Bose gas with arbitrary repulsive interaction potential of finite
scattering length. For this purpose, we prove bounds on both the eigenvalues and
corresponding eigenfunctions of three-dimensional bosons in strongly elongated
traps and relate them to the corresponding quantities in the Lieb-Liniger model.
In particular, if both the scattering length a and the radius r of the cylindrical
trap go to zero, the Lieb-Liniger model with coupling constant g ∼ a/r 2 is derived.
Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply
to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼
r -2 above the ground state energy.
author:
- first_name: Robert
full_name: Robert Seiringer
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Seiringer R, Yin J. The Lieb-Liniger model as a limit of dilute bosons in three
dimensions. *Communications in Mathematical Physics*. 2008;284(2):459-479.
doi:10.1007/s00220-008-0521-6
apa: Seiringer, R., & Yin, J. (2008). The Lieb-Liniger model as a limit of dilute
bosons in three dimensions. *Communications in Mathematical Physics*. Springer.
https://doi.org/10.1007/s00220-008-0521-6
chicago: Seiringer, Robert, and Jun Yin. “The Lieb-Liniger Model as a Limit of Dilute
Bosons in Three Dimensions.” *Communications in Mathematical Physics*. Springer,
2008. https://doi.org/10.1007/s00220-008-0521-6.
ieee: R. Seiringer and J. Yin, “The Lieb-Liniger model as a limit of dilute bosons
in three dimensions,” *Communications in Mathematical Physics*, vol. 284,
no. 2. Springer, pp. 459–479, 2008.
ista: Seiringer R, Yin J. 2008. The Lieb-Liniger model as a limit of dilute bosons
in three dimensions. Communications in Mathematical Physics. 284(2), 459–479.
mla: Seiringer, Robert, and Jun Yin. “The Lieb-Liniger Model as a Limit of Dilute
Bosons in Three Dimensions.” *Communications in Mathematical Physics*, vol.
284, no. 2, Springer, 2008, pp. 459–79, doi:10.1007/s00220-008-0521-6.
short: R. Seiringer, J. Yin, Communications in Mathematical Physics 284 (2008) 459–479.
date_created: 2018-12-11T11:57:21Z
date_published: 2008-12-01T00:00:00Z
date_updated: 2021-01-12T06:57:08Z
day: '01'
doi: 10.1007/s00220-008-0521-6
extern: 1
intvolume: ' 284'
issue: '2'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/0709.4022
month: '12'
oa: 1
page: 459 - 479
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4544'
quality_controlled: 0
status: public
title: The Lieb-Liniger model as a limit of dilute bosons in three dimensions
type: journal_article
volume: 284
year: '2008'
...