---
OA_place: publisher
OA_type: hybrid
_id: '19403'
abstract:
- lang: eng
text: We study the BCS critical temperature on half-spaces in dimensions d =1, 2,
3 with Dirichlet or Neumann boundary conditions. We prove that the critical temperature
on a half-space is strictly higher than on Rd, at least at weak coupling in d
= 1, 2 and weak coupling and small chemical potential in d = 3. Furthermore, we
show that the relative shift in critical temperature vanishes in the weak coupling
limit.
acknowledgement: Open access funding provided by Institute of Science and Technology
(IST Austria). Financial support by the Austrian Science Fund (FWF) through project
number I 6427-N (as part of the SFB/TRR 352) is gratefully acknowledged.
article_number: '20'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Barbara
full_name: Roos, Barbara
id: 5DA90512-D80F-11E9-8994-2E2EE6697425
last_name: Roos
orcid: 0000-0002-9071-5880
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Roos B, Seiringer R. BCS critical temperature on half-spaces. Archive for
Rational Mechanics and Analysis. 2025;249. doi:10.1007/s00205-025-02088-x
apa: Roos, B., & Seiringer, R. (2025). BCS critical temperature on half-spaces.
Archive for Rational Mechanics and Analysis. Springer Nature. https://doi.org/10.1007/s00205-025-02088-x
chicago: Roos, Barbara, and Robert Seiringer. “BCS Critical Temperature on Half-Spaces.”
Archive for Rational Mechanics and Analysis. Springer Nature, 2025. https://doi.org/10.1007/s00205-025-02088-x.
ieee: B. Roos and R. Seiringer, “BCS critical temperature on half-spaces,” Archive
for Rational Mechanics and Analysis, vol. 249. Springer Nature, 2025.
ista: Roos B, Seiringer R. 2025. BCS critical temperature on half-spaces. Archive
for Rational Mechanics and Analysis. 249, 20.
mla: Roos, Barbara, and Robert Seiringer. “BCS Critical Temperature on Half-Spaces.”
Archive for Rational Mechanics and Analysis, vol. 249, 20, Springer Nature,
2025, doi:10.1007/s00205-025-02088-x.
short: B. Roos, R. Seiringer, Archive for Rational Mechanics and Analysis 249 (2025).
corr_author: '1'
date_created: 2025-03-16T23:01:24Z
date_published: 2025-04-01T00:00:00Z
date_updated: 2025-09-30T11:01:08Z
day: '01'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1007/s00205-025-02088-x
external_id:
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- '2306.05824'
isi:
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pmid:
- '40041541'
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name: Mathematical Challenges in BCS Theory of Superconductivity
publication: Archive for Rational Mechanics and Analysis
publication_identifier:
eissn:
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issn:
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publisher: Springer Nature
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title: BCS critical temperature on half-spaces
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abstract:
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text: "We consider the BCS energy gap „.T / (essentially given by „.T / \x19 \x81.T;
p\x16/,\r\nthe BCS order parameter) at all temperatures 0 \x14 T \x14 Tc up to
the critical one, Tc, and show\r\nthat, in the limit of weak coupling, the ratio
„.T /=Tc is given by a universal function of the relative temperature T =Tc. On
the one hand, this recovers a recent result by Langmann and Triola\r\n[Phys. Rev.
B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors
for temperatures bounded uniformly away from Tc. On the other hand, our result
lifts these\r\nrestrictions, as we consider arbitrary spatial dimensions d 2 ¹1;
2; 3º, discuss superconductors\r\nwith non-zero angular momentum (primarily in
two dimensions), and treat the perhaps physically most interesting (due to the
occurrence of the superconducting phase transition) regime of\r\ntemperatures
close to Tc.\r\n\r\n\r\n ."
acknowledgement: "We thank Andreas Deuchert, Christian Hainzl, Edwin Langmann, Marius
Lemm, Robert Seiringer, and Jan Philip Solovej for helpful discussions,\r\nand Edwin
Langmann and Robert Seiringer for valuable comments on an earlier version of the
manuscript.\r\nFunding. Joscha Henheik gratefully acknowledges partial financial
support by the\r\nERC Advanced Grant “RMTBeyond” No. 101020331. Asbjørn Bækgaard
Lauritsen\r\ngratefully acknowledges partial financial support by the Austrian Science
Fund (FWF)\r\nthrough grant DOI 10.55776/I6427 (as part of the SFB/TRR 352).\r\n"
article_processing_charge: No
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arxiv: 1
author:
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full_name: Henheik, Sven Joscha
id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
last_name: Henheik
orcid: 0000-0003-1106-327X
- first_name: Asbjørn Bækgaard
full_name: Lauritsen, Asbjørn Bækgaard
id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
last_name: Lauritsen
orcid: 0000-0003-4476-2288
citation:
ama: Henheik SJ, Lauritsen AB. Universal behavior of the BCS energy gap. Journal
of Spectral Theory. 2025;15(1):305–352. doi:10.4171/JST/540
apa: Henheik, S. J., & Lauritsen, A. B. (2025). Universal behavior of the BCS
energy gap. Journal of Spectral Theory. EMS Press. https://doi.org/10.4171/JST/540
chicago: Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior
of the BCS Energy Gap.” Journal of Spectral Theory. EMS Press, 2025. https://doi.org/10.4171/JST/540.
ieee: S. J. Henheik and A. B. Lauritsen, “Universal behavior of the BCS energy gap,”
Journal of Spectral Theory, vol. 15, no. 1. EMS Press, pp. 305–352, 2025.
ista: Henheik SJ, Lauritsen AB. 2025. Universal behavior of the BCS energy gap.
Journal of Spectral Theory. 15(1), 305–352.
mla: Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior of
the BCS Energy Gap.” Journal of Spectral Theory, vol. 15, no. 1, EMS Press,
2025, pp. 305–352, doi:10.4171/JST/540.
short: S.J. Henheik, A.B. Lauritsen, Journal of Spectral Theory 15 (2025) 305–352.
corr_author: '1'
date_created: 2025-04-11T09:19:28Z
date_published: 2025-01-09T00:00:00Z
date_updated: 2026-04-07T12:37:11Z
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ddc:
- '500'
department:
- _id: LaEr
- _id: RoSe
doi: 10.4171/JST/540
ec_funded: 1
external_id:
arxiv:
- '2312.11310'
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- '001438931600009'
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grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
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grant_number: I06427
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- 1664-0403
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title: Universal behavior of the BCS energy gap
tmp:
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legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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...
---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '19628'
abstract:
- lang: eng
text: We consider the critical temperature for superconductivity, defined via the
linear BCS equation. We prove that at weak coupling the critical temperature for
a sample confined to a quadrant in two dimensions is strictly larger than the
one for a half-space, which in turn is strictly larger than the one for R^2.
Furthermore, we prove that the relative difference of the critical temperatures
vanishes in the weak coupling limit.
article_number: e71
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Barbara
full_name: Roos, Barbara
id: 5DA90512-D80F-11E9-8994-2E2EE6697425
last_name: Roos
orcid: 0000-0002-9071-5880
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Roos B, Seiringer R. Enhanced superconductivity at a corner for the linear
BCS equation. Forum of Mathematics, Sigma. 2025;13. doi:10.1017/fms.2024.145
apa: Roos, B., & Seiringer, R. (2025). Enhanced superconductivity at a corner
for the linear BCS equation. Forum of Mathematics, Sigma. Cambridge University
Press. https://doi.org/10.1017/fms.2024.145
chicago: Roos, Barbara, and Robert Seiringer. “Enhanced Superconductivity at a Corner
for the Linear BCS Equation.” Forum of Mathematics, Sigma. Cambridge University
Press, 2025. https://doi.org/10.1017/fms.2024.145.
ieee: B. Roos and R. Seiringer, “Enhanced superconductivity at a corner for the
linear BCS equation,” Forum of Mathematics, Sigma, vol. 13. Cambridge University
Press, 2025.
ista: Roos B, Seiringer R. 2025. Enhanced superconductivity at a corner for the
linear BCS equation. Forum of Mathematics, Sigma. 13, e71.
mla: Roos, Barbara, and Robert Seiringer. “Enhanced Superconductivity at a Corner
for the Linear BCS Equation.” Forum of Mathematics, Sigma, vol. 13, e71,
Cambridge University Press, 2025, doi:10.1017/fms.2024.145.
short: B. Roos, R. Seiringer, Forum of Mathematics, Sigma 13 (2025).
corr_author: '1'
date_created: 2025-04-27T22:02:14Z
date_published: 2025-04-14T00:00:00Z
date_updated: 2025-09-30T12:20:22Z
day: '14'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1017/fms.2024.145
external_id:
arxiv:
- '2308.07115'
isi:
- '001465985500001'
file:
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file_name: 2025_ForumMathSigma_Roos.pdf
file_size: 631645
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file_date_updated: 2025-05-05T09:41:42Z
has_accepted_license: '1'
intvolume: ' 13'
isi: 1
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: bda63fe5-d553-11ed-ba76-a16e3d2f256b
grant_number: I06427
name: Mathematical Challenges in BCS Theory of Superconductivity
publication: Forum of Mathematics, Sigma
publication_identifier:
eissn:
- 2050-5094
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Enhanced superconductivity at a corner for the linear BCS equation
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 13
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '17240'
abstract:
- lang: eng
text: We prove an upper bound on the energy density of the dilute spin-\(\frac {1}{2}\)
Fermi gas capturing the leading correction to the kinetic energy\(8\pi a\rho _\uparrow\rho
_\downarrow\) with an error of size smaller than\(a\rho^{2}(a^ 3\rho)^{1/3-\varepsilon}\)
for any\(\varepsilon> 0\), where a denotes the scattering length of the interaction.
The result is valid for a large class of interactions including interactions with
a hard core. A central ingredient in the proof is a rigorous version of a fermionic
cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys
A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3).
acknowledgement: "Open access funding provided by Institute of Science and Technology
(IST Austria).\r\nWe thank Alessandro Giuliani and Robert Seiringer for helpful
discussions and Robert Seiringer for his comments on the manuscript. Financial support
by the Austrian Science Fund (FWF) through Grant https://doi.org/10.55776/I6427
(as part of the SFB/TRR 352) is gratefully acknowledged."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Asbjørn Bækgaard
full_name: Lauritsen, Asbjørn Bækgaard
id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
last_name: Lauritsen
orcid: 0000-0003-4476-2288
citation:
ama: Lauritsen AB. Almost optimal upper bound for the ground state energy of a dilute
Fermi gas via cluster expansion. Annales Henri Poincare. 2025;26:203-243.
doi:10.1007/s00023-024-01450-1
apa: Lauritsen, A. B. (2025). Almost optimal upper bound for the ground state energy
of a dilute Fermi gas via cluster expansion. Annales Henri Poincare. Springer
Nature. https://doi.org/10.1007/s00023-024-01450-1
chicago: Lauritsen, Asbjørn Bækgaard. “Almost Optimal Upper Bound for the Ground
State Energy of a Dilute Fermi Gas via Cluster Expansion.” Annales Henri Poincare.
Springer Nature, 2025. https://doi.org/10.1007/s00023-024-01450-1.
ieee: A. B. Lauritsen, “Almost optimal upper bound for the ground state energy of
a dilute Fermi gas via cluster expansion,” Annales Henri Poincare, vol.
26. Springer Nature, pp. 203–243, 2025.
ista: Lauritsen AB. 2025. Almost optimal upper bound for the ground state energy
of a dilute Fermi gas via cluster expansion. Annales Henri Poincare. 26, 203–243.
mla: Lauritsen, Asbjørn Bækgaard. “Almost Optimal Upper Bound for the Ground State
Energy of a Dilute Fermi Gas via Cluster Expansion.” Annales Henri Poincare,
vol. 26, Springer Nature, 2025, pp. 203–43, doi:10.1007/s00023-024-01450-1.
short: A.B. Lauritsen, Annales Henri Poincare 26 (2025) 203–243.
corr_author: '1'
date_created: 2024-07-14T22:01:12Z
date_published: 2025-01-01T00:00:00Z
date_updated: 2026-04-07T13:01:40Z
day: '01'
ddc:
- '510'
department:
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doi: 10.1007/s00023-024-01450-1
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publication: Annales Henri Poincare
publication_identifier:
issn:
- 1424-0637
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
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title: Almost optimal upper bound for the ground state energy of a dilute Fermi gas
via cluster expansion
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type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
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...
---
OA_place: publisher
OA_type: hybrid
_id: '14542'
abstract:
- lang: eng
text: "It is a remarkable property of BCS theory that the ratio of the energy gap
at zero temperature Ξ\r\n and the critical temperature Tc is (approximately) given
by a universal constant, independent of the microscopic details of the fermionic
interaction. This universality has rigorously been proven quite recently in three
spatial dimensions and three different limiting regimes: weak coupling, low density
and high density. The goal of this short note is to extend the universal behavior
to lower dimensions d=1,2 and give an exemplary proof in the weak coupling limit."
acknowledgement: We thank Robert Seiringer for comments on the paper. J. H. gratefully
acknowledges partial financial support by the ERC Advanced Grant “RMTBeyond”No.
101020331.This research was funded in part by the Austrian Science Fund (FWF) grantnumber
I6427.
article_number: '2360005 '
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Sven Joscha
full_name: Henheik, Sven Joscha
id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
last_name: Henheik
orcid: 0000-0003-1106-327X
- first_name: Asbjørn Bækgaard
full_name: Lauritsen, Asbjørn Bækgaard
id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
last_name: Lauritsen
orcid: 0000-0003-4476-2288
- first_name: Barbara
full_name: Roos, Barbara
id: 5DA90512-D80F-11E9-8994-2E2EE6697425
last_name: Roos
orcid: 0000-0002-9071-5880
citation:
ama: Henheik SJ, Lauritsen AB, Roos B. Universality in low-dimensional BCS theory.
Reviews in Mathematical Physics. 2024;36(9). doi:10.1142/s0129055x2360005x
apa: Henheik, S. J., Lauritsen, A. B., & Roos, B. (2024). Universality in low-dimensional
BCS theory. Reviews in Mathematical Physics. World Scientific Publishing.
https://doi.org/10.1142/s0129055x2360005x
chicago: Henheik, Sven Joscha, Asbjørn Bækgaard Lauritsen, and Barbara Roos. “Universality
in Low-Dimensional BCS Theory.” Reviews in Mathematical Physics. World
Scientific Publishing, 2024. https://doi.org/10.1142/s0129055x2360005x.
ieee: S. J. Henheik, A. B. Lauritsen, and B. Roos, “Universality in low-dimensional
BCS theory,” Reviews in Mathematical Physics, vol. 36, no. 9. World Scientific
Publishing, 2024.
ista: Henheik SJ, Lauritsen AB, Roos B. 2024. Universality in low-dimensional BCS
theory. Reviews in Mathematical Physics. 36(9), 2360005.
mla: Henheik, Sven Joscha, et al. “Universality in Low-Dimensional BCS Theory.”
Reviews in Mathematical Physics, vol. 36, no. 9, 2360005, World Scientific
Publishing, 2024, doi:10.1142/s0129055x2360005x.
short: S.J. Henheik, A.B. Lauritsen, B. Roos, Reviews in Mathematical Physics 36
(2024).
corr_author: '1'
date_created: 2023-11-15T23:48:14Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2026-04-07T13:01:40Z
day: '01'
ddc:
- '510'
department:
- _id: GradSch
- _id: LaEr
- _id: RoSe
doi: 10.1142/s0129055x2360005x
ec_funded: 1
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- '001099640300002'
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- iso: eng
month: '10'
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oa_version: Published Version
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
- _id: bda63fe5-d553-11ed-ba76-a16e3d2f256b
grant_number: I06427
name: Mathematical Challenges in BCS Theory of Superconductivity
publication: Reviews in Mathematical Physics
publication_identifier:
eissn:
- 1793-6659
issn:
- 0129-055X
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
related_material:
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- id: '19540'
relation: dissertation_contains
status: public
- id: '18135'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Universality in low-dimensional BCS theory
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 36
year: '2024'
...
---
_id: '14931'
abstract:
- lang: eng
text: We prove an upper bound on the ground state energy of the dilute spin-polarized
Fermi gas capturing the leading correction to the kinetic energy resulting from
repulsive interactions. One of the main ingredients in the proof is a rigorous
implementation of the fermionic cluster expansion of Gaudin et al. (1971) [15].
acknowledgement: A.B.L. would like to thank Johannes Agerskov and Jan Philip Solovej
for valuable discussions. We thank Alessandro Giuliani for helpful discussions and
for pointing out the reference [18]. Funding from the European Union's Horizon 2020
research and innovation programme under the ERC grant agreement No 694227 is acknowledged.
Financial support by the Austrian Science Fund (FWF) through project number I 6427-N
(as part of the SFB/TRR 352) is gratefully acknowledged.
article_number: '110320'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Asbjørn Bækgaard
full_name: Lauritsen, Asbjørn Bækgaard
id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
last_name: Lauritsen
orcid: 0000-0003-4476-2288
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: 'Lauritsen AB, Seiringer R. Ground state energy of the dilute spin-polarized
Fermi gas: Upper bound via cluster expansion. Journal of Functional Analysis.
2024;286(7). doi:10.1016/j.jfa.2024.110320'
apa: 'Lauritsen, A. B., & Seiringer, R. (2024). Ground state energy of the dilute
spin-polarized Fermi gas: Upper bound via cluster expansion. Journal of Functional
Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2024.110320'
chicago: 'Lauritsen, Asbjørn Bækgaard, and Robert Seiringer. “Ground State Energy
of the Dilute Spin-Polarized Fermi Gas: Upper Bound via Cluster Expansion.” Journal
of Functional Analysis. Elsevier, 2024. https://doi.org/10.1016/j.jfa.2024.110320.'
ieee: 'A. B. Lauritsen and R. Seiringer, “Ground state energy of the dilute spin-polarized
Fermi gas: Upper bound via cluster expansion,” Journal of Functional Analysis,
vol. 286, no. 7. Elsevier, 2024.'
ista: 'Lauritsen AB, Seiringer R. 2024. Ground state energy of the dilute spin-polarized
Fermi gas: Upper bound via cluster expansion. Journal of Functional Analysis.
286(7), 110320.'
mla: 'Lauritsen, Asbjørn Bækgaard, and Robert Seiringer. “Ground State Energy of
the Dilute Spin-Polarized Fermi Gas: Upper Bound via Cluster Expansion.” Journal
of Functional Analysis, vol. 286, no. 7, 110320, Elsevier, 2024, doi:10.1016/j.jfa.2024.110320.'
short: A.B. Lauritsen, R. Seiringer, Journal of Functional Analysis 286 (2024).
corr_author: '1'
date_created: 2024-02-04T23:00:53Z
date_published: 2024-04-01T00:00:00Z
date_updated: 2026-04-16T08:17:56Z
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title: 'Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via
cluster expansion'
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legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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_id: '18107'
abstract:
- lang: eng
text: We consider a dilute fully spin-polarized Fermi gas at positive temperature
in dimensions d∈{1,2,3} . We show that the pressure of the interacting gas is
bounded from below by that of the free gas plus, to leading order, an explicit
term of order adρ2+2/d, where a is the p-wave scattering length of the repulsive
interaction and ρ is the particle density. The results are valid for a wide
range of repulsive interactions, including that of a hard core, and uniform in
temperatures at most of the order of the Fermi temperature. A central ingredient
in the proof is a rigorous implementation of the fermionic cluster expansion of
Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).
acknowledgement: 'Financial support by the Austrian Science Fund (FWF) through grant
DOI: 10.55776/I6427 (as part of the SFB/TRR 352) is gratefully acknowledged.'
article_number: e78
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Asbjørn Bækgaard
full_name: Lauritsen, Asbjørn Bækgaard
id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
last_name: Lauritsen
orcid: 0000-0003-4476-2288
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: 'Lauritsen AB, Seiringer R. Pressure of a dilute spin-polarized Fermi gas:
Lower bound. Forum of Mathematics, Sigma. 2024;12. doi:10.1017/fms.2024.56'
apa: 'Lauritsen, A. B., & Seiringer, R. (2024). Pressure of a dilute spin-polarized
Fermi gas: Lower bound. Forum of Mathematics, Sigma. Cambridge University
Press. https://doi.org/10.1017/fms.2024.56'
chicago: 'Lauritsen, Asbjørn Bækgaard, and Robert Seiringer. “Pressure of a Dilute
Spin-Polarized Fermi Gas: Lower Bound.” Forum of Mathematics, Sigma. Cambridge
University Press, 2024. https://doi.org/10.1017/fms.2024.56.'
ieee: 'A. B. Lauritsen and R. Seiringer, “Pressure of a dilute spin-polarized Fermi
gas: Lower bound,” Forum of Mathematics, Sigma, vol. 12. Cambridge University
Press, 2024.'
ista: 'Lauritsen AB, Seiringer R. 2024. Pressure of a dilute spin-polarized Fermi
gas: Lower bound. Forum of Mathematics, Sigma. 12, e78.'
mla: 'Lauritsen, Asbjørn Bækgaard, and Robert Seiringer. “Pressure of a Dilute Spin-Polarized
Fermi Gas: Lower Bound.” Forum of Mathematics, Sigma, vol. 12, e78, Cambridge
University Press, 2024, doi:10.1017/fms.2024.56.'
short: A.B. Lauritsen, R. Seiringer, Forum of Mathematics, Sigma 12 (2024).
corr_author: '1'
date_created: 2024-09-20T12:25:25Z
date_published: 2024-09-09T00:00:00Z
date_updated: 2026-04-07T13:01:40Z
day: '09'
ddc:
- '510'
department:
- _id: GradSch
- _id: RoSe
doi: 10.1017/fms.2024.56
external_id:
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title: 'Pressure of a dilute spin-polarized Fermi gas: Lower bound'
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user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
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abstract:
- lang: eng
text: "This thesis consists of two separate parts. In the first part we consider
a dilute Fermi gas interacting through a repulsive interaction in dimensions $d=1,2,3$.
Our focus is mostly on the physically most relevant dimension $d=3$ \r\nand the
setting of a spin-polarized (equivalently spinless) gas, where the Pauli exclusion
principle plays a key role. We show that, at zero temperature, the ground state
energy density of the interacting spin-polarized gas differs (to leading order)
from that of the free (i.e. non-interacting) gas by a term of order $a_p^d\\rho^{2+2/d}$
\ with $a_p$ the $p$-wave scattering length of the repulsive interaction and $\\rho$
the density. Further, we extend this to positive temperature and show that the
pressure of an interacting spin-polarized gas differs from that of the free gas
by a now temperature dependent term, again of order $a_p^d\\rho^{2+2/d}$. Lastly,
we consider the setting of a spin-$\\frac{1}{2}$ Fermi gas in $d=3$ dimensions
and show that here, as an upper bound, the ground state energy density differs
from that of the free system by a term of order $a_s \\rho^2$ with an error smaller
than $a_s \\rho^2 (a_s\\rho^{1/3})^{1-\\eps}$ for any $\\eps > 0$, where $a_s$
is the $s$-wave scattering length of the repulsive interaction. \r\n\r\nThese
asymptotic formulas complement the similar formulas in the literature for the
dilute Bose and spin-$\\frac{1}{2}$ Fermi gas, where the ground state energies
or pressures differ from that of the corresponding free systems by a term of order
$a_s \\rho^2$ in dimension $d=3$. In the spin-polarized setting, the corrections,
of order $a_p^3\\rho^{8/3}$ in dimension $d=3$, are thus much smaller and requires
a more delicate analysis.\r\n\r\nIn the second part of the thesis we consider
the Bardeen--Cooper--Schrieffer (BCS) theory of superconductivity and in particular
its associated critical temperature and energy gap. We prove that the ratio of
the zero-temperature energy gap and critical temperature $\\Xi(T=0)/T_c$ approaches
a universal constant $\\pi e^{-\\gamma}\\approx 1.76$ in both the limit of high
density in dimension $d=3$ and in the limit of weak coupling in dimensions $d=1,2$.
This complements the proofs in the literature of this universal behaviour in the
limit of weak coupling or low density in dimension $d=3$. Secondly, we prove that
the ratio of the energy gap at positive temperature and critical temperature $\\Xi(T)/T_c$
approaches a universal function of the relative temperature $T/T_c$ in the limit
of weak coupling in dimensions $d=1,2,3$."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Asbjørn Bækgaard
full_name: Lauritsen, Asbjørn Bækgaard
id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1
last_name: Lauritsen
orcid: 0000-0003-4476-2288
citation:
ama: Lauritsen AB. Energies of dilute Fermi gases and universalities in BCS theory.
2024. doi:10.15479/at:ista:18135
apa: Lauritsen, A. B. (2024). Energies of dilute Fermi gases and universalities
in BCS theory. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:18135
chicago: Lauritsen, Asbjørn Bækgaard. “Energies of Dilute Fermi Gases and Universalities
in BCS Theory.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:18135.
ieee: A. B. Lauritsen, “Energies of dilute Fermi gases and universalities in BCS
theory,” Institute of Science and Technology Austria, 2024.
ista: Lauritsen AB. 2024. Energies of dilute Fermi gases and universalities in BCS
theory. Institute of Science and Technology Austria.
mla: Lauritsen, Asbjørn Bækgaard. Energies of Dilute Fermi Gases and Universalities
in BCS Theory. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:18135.
short: A.B. Lauritsen, Energies of Dilute Fermi Gases and Universalities in BCS
Theory, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-09-24T10:56:25Z
date_published: 2024-09-23T00:00:00Z
date_updated: 2026-04-16T08:17:55Z
day: '23'
ddc:
- '515'
- '539'
degree_awarded: PhD
department:
- _id: GradSch
- _id: RoSe
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grant_number: I06427
name: Mathematical Challenges in BCS Theory of Superconductivity
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
publication_identifier:
isbn:
- 978-3-99078-042-8
issn:
- 2663-337X
publication_status: published
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supervisor:
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
title: Energies of dilute Fermi gases and universalities in BCS theory
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abstract:
- lang: eng
text: "Superconductivity has many important applications ranging from levitating
trains over qubits to MRI scanners. The phenomenon is successfully modeled by
Bardeen-Cooper-Schrieffer (BCS) theory. From a mathematical perspective, BCS theory
has been studied extensively for systems without boundary. However, little is
known in the presence of boundaries. With the help of numerical methods physicists
observed that the critical temperature may increase in the presence of a boundary.
The goal of this thesis is to understand the influence of boundaries on the critical
temperature in BCS theory and to give a first rigorous justification of these
observations. On the way, we also study two-body Schrödinger operators on domains
with boundaries and prove additional results for superconductors without boundary.\r\n\r\nBCS
theory is based on a non-linear functional, where the minimizer indicates whether
the system is superconducting or in the normal, non-superconducting state. By
considering the Hessian of the BCS functional at the normal state, one can analyze
whether the normal state is possibly a minimum of the BCS functional and estimate
the critical temperature. The Hessian turns out to be a linear operator resembling
a Schrödinger operator for two interacting particles, but with more complicated
kinetic energy. As a first step, we study the two-body Schrödinger operator in
the presence of boundaries.\r\nFor Neumann boundary conditions, we prove that
the addition of a boundary can create new eigenvalues, which correspond to the
two particles forming a bound state close to the boundary.\r\n\r\nSecond, we need
to understand superconductivity in the translation invariant setting. While in
three dimensions this has been extensively studied, there is no mathematical literature
for the one and two dimensional cases. In dimensions one and two, we compute the
weak coupling asymptotics of the critical temperature and the energy gap in the
translation invariant setting. We also prove that their ratio is independent of
the microscopic details of the model in the weak coupling limit; this property
is referred to as universality.\r\n\r\nIn the third part, we study the critical
temperature of superconductors in the presence of boundaries. We start by considering
the one-dimensional case of a half-line with contact interaction. Then, we generalize
the results to generic interactions and half-spaces in one, two and three dimensions.
Finally, we compare the critical temperature of a quarter space in two dimensions
to the critical temperatures of a half-space and of the full space."
alternative_title:
- ISTA Thesis
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author:
- first_name: Barbara
full_name: Roos, Barbara
id: 5DA90512-D80F-11E9-8994-2E2EE6697425
last_name: Roos
orcid: 0000-0002-9071-5880
citation:
ama: Roos B. Boundary superconductivity in BCS theory. 2023. doi:10.15479/at:ista:14374
apa: Roos, B. (2023). Boundary superconductivity in BCS theory. Institute
of Science and Technology Austria. https://doi.org/10.15479/at:ista:14374
chicago: Roos, Barbara. “Boundary Superconductivity in BCS Theory.” Institute of
Science and Technology Austria, 2023. https://doi.org/10.15479/at:ista:14374.
ieee: B. Roos, “Boundary superconductivity in BCS theory,” Institute of Science
and Technology Austria, 2023.
ista: Roos B. 2023. Boundary superconductivity in BCS theory. Institute of Science
and Technology Austria.
mla: Roos, Barbara. Boundary Superconductivity in BCS Theory. Institute of
Science and Technology Austria, 2023, doi:10.15479/at:ista:14374.
short: B. Roos, Boundary Superconductivity in BCS Theory, Institute of Science and
Technology Austria, 2023.
corr_author: '1'
date_created: 2023-09-28T14:23:04Z
date_published: 2023-09-30T00:00:00Z
date_updated: 2026-04-07T13:27:39Z
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name: Analysis of quantum many-body systems
- _id: bda63fe5-d553-11ed-ba76-a16e3d2f256b
grant_number: I06427
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