---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
_id: '19548'
abstract:
- lang: eng
  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
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
citation:
  ama: Henheik SJ, Lauritsen AB. Universal behavior of the BCS energy gap. <i>Journal
    of Spectral Theory</i>. 2025;15(1):305–352. doi:<a href="https://doi.org/10.4171/JST/540">10.4171/JST/540</a>
  apa: Henheik, S. J., &#38; Lauritsen, A. B. (2025). Universal behavior of the BCS
    energy gap. <i>Journal of Spectral Theory</i>. EMS Press. <a href="https://doi.org/10.4171/JST/540">https://doi.org/10.4171/JST/540</a>
  chicago: Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior
    of the BCS Energy Gap.” <i>Journal of Spectral Theory</i>. EMS Press, 2025. <a
    href="https://doi.org/10.4171/JST/540">https://doi.org/10.4171/JST/540</a>.
  ieee: S. J. Henheik and A. B. Lauritsen, “Universal behavior of the BCS energy gap,”
    <i>Journal of Spectral Theory</i>, 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.” <i>Journal of Spectral Theory</i>, vol. 15, no. 1, EMS Press,
    2025, pp. 305–352, doi:<a href="https://doi.org/10.4171/JST/540">10.4171/JST/540</a>.
  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
day: '09'
ddc:
- '500'
department:
- _id: LaEr
- _id: RoSe
doi: 10.4171/JST/540
ec_funded: 1
external_id:
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  - '2312.11310'
  isi:
  - '001438931600009'
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intvolume: '        15'
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issue: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 305–352
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: Journal of Spectral Theory
publication_identifier:
  eissn:
  - 1664-0403
publication_status: published
publisher: EMS Press
quality_controlled: '1'
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title: Universal behavior of the BCS energy gap
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---
DOAJ_listed: '1'
OA_place: publisher
OA_type: diamond
_id: '14662'
abstract:
- lang: eng
  text: We consider a class of polaron models, including the Fröhlich model, at zero
    total momentum, and show that at sufficiently weak coupling there are no excited
    eigenvalues below the essential spectrum.
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Robert
  full_name: Seiringer, Robert
  id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
  last_name: Seiringer
  orcid: 0000-0002-6781-0521
citation:
  ama: Seiringer R. Absence of excited eigenvalues for Fröhlich type polaron models
    at weak coupling. <i>Journal of Spectral Theory</i>. 2023;13(3):1045-1055. doi:<a
    href="https://doi.org/10.4171/JST/469">10.4171/JST/469</a>
  apa: Seiringer, R. (2023). Absence of excited eigenvalues for Fröhlich type polaron
    models at weak coupling. <i>Journal of Spectral Theory</i>. EMS Press. <a href="https://doi.org/10.4171/JST/469">https://doi.org/10.4171/JST/469</a>
  chicago: Seiringer, Robert. “Absence of Excited Eigenvalues for Fröhlich Type Polaron
    Models at Weak Coupling.” <i>Journal of Spectral Theory</i>. EMS Press, 2023.
    <a href="https://doi.org/10.4171/JST/469">https://doi.org/10.4171/JST/469</a>.
  ieee: R. Seiringer, “Absence of excited eigenvalues for Fröhlich type polaron models
    at weak coupling,” <i>Journal of Spectral Theory</i>, vol. 13, no. 3. EMS Press,
    pp. 1045–1055, 2023.
  ista: Seiringer R. 2023. Absence of excited eigenvalues for Fröhlich type polaron
    models at weak coupling. Journal of Spectral Theory. 13(3), 1045–1055.
  mla: Seiringer, Robert. “Absence of Excited Eigenvalues for Fröhlich Type Polaron
    Models at Weak Coupling.” <i>Journal of Spectral Theory</i>, vol. 13, no. 3, EMS
    Press, 2023, pp. 1045–55, doi:<a href="https://doi.org/10.4171/JST/469">10.4171/JST/469</a>.
  short: R. Seiringer, Journal of Spectral Theory 13 (2023) 1045–1055.
corr_author: '1'
date_created: 2023-12-10T23:00:59Z
date_published: 2023-11-25T00:00:00Z
date_updated: 2025-09-09T13:40:33Z
day: '25'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.4171/JST/469
external_id:
  arxiv:
  - '2210.17123'
  isi:
  - '001127467700008'
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intvolume: '        13'
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issue: '3'
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- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 1045-1055
publication: Journal of Spectral Theory
publication_identifier:
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  - 1664-0403
  issn:
  - 1664-039X
publication_status: published
publisher: EMS Press
quality_controlled: '1'
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title: Absence of excited eigenvalues for Fröhlich type polaron models at weak coupling
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type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
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year: '2023'
...
---
_id: '13207'
abstract:
- lang: eng
  text: We consider the linear BCS equation, determining the BCS critical temperature,
    in the presence of a boundary, where Dirichlet boundary conditions are imposed.
    In the one-dimensional case with point interactions, we prove that the critical
    temperature is strictly larger than the bulk value, at least at weak coupling.
    In particular, the Cooper-pair wave function localizes near the boundary, an effect
    that cannot be modeled by effective Neumann boundary conditions on the order parameter
    as often imposed in Ginzburg–Landau theory. We also show that the relative shift
    in critical temperature vanishes if the coupling constant either goes to zero
    or to infinity.
acknowledgement: We thank Egor Babaev for encouraging us to study this problem, and
  Rupert Frank for many fruitful discussions. scussions. Funding. Funding from the
  European Union’s Horizon 2020 research and innovation programme under the ERC grant
  agreement No. 694227 (Barbara Roos and Robert Seiringer) is gratefully acknowledged.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Christian
  full_name: Hainzl, Christian
  last_name: Hainzl
- 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: Hainzl C, Roos B, Seiringer R. Boundary superconductivity in the BCS model.
    <i>Journal of Spectral Theory</i>. 2023;12(4):1507–1540. doi:<a href="https://doi.org/10.4171/JST/439">10.4171/JST/439</a>
  apa: Hainzl, C., Roos, B., &#38; Seiringer, R. (2023). Boundary superconductivity
    in the BCS model. <i>Journal of Spectral Theory</i>. EMS Press. <a href="https://doi.org/10.4171/JST/439">https://doi.org/10.4171/JST/439</a>
  chicago: Hainzl, Christian, Barbara Roos, and Robert Seiringer. “Boundary Superconductivity
    in the BCS Model.” <i>Journal of Spectral Theory</i>. EMS Press, 2023. <a href="https://doi.org/10.4171/JST/439">https://doi.org/10.4171/JST/439</a>.
  ieee: C. Hainzl, B. Roos, and R. Seiringer, “Boundary superconductivity in the BCS
    model,” <i>Journal of Spectral Theory</i>, vol. 12, no. 4. EMS Press, pp. 1507–1540,
    2023.
  ista: Hainzl C, Roos B, Seiringer R. 2023. Boundary superconductivity in the BCS
    model. Journal of Spectral Theory. 12(4), 1507–1540.
  mla: Hainzl, Christian, et al. “Boundary Superconductivity in the BCS Model.” <i>Journal
    of Spectral Theory</i>, vol. 12, no. 4, EMS Press, 2023, pp. 1507–1540, doi:<a
    href="https://doi.org/10.4171/JST/439">10.4171/JST/439</a>.
  short: C. Hainzl, B. Roos, R. Seiringer, Journal of Spectral Theory 12 (2023) 1507–1540.
corr_author: '1'
date_created: 2023-07-10T16:35:45Z
date_published: 2023-05-18T00:00:00Z
date_updated: 2026-04-07T13:27:39Z
day: '18'
ddc:
- '530'
department:
- _id: GradSch
- _id: RoSe
doi: 10.4171/JST/439
ec_funded: 1
external_id:
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  - '2201.08090'
  isi:
  - '000997933500008'
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month: '05'
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oa_version: Published Version
page: 1507–1540
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '694227'
  name: Analysis of quantum many-body systems
publication: Journal of Spectral Theory
publication_identifier:
  eissn:
  - 1664-0403
  issn:
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publication_status: published
publisher: EMS Press
quality_controlled: '1'
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...