[{"oa":1,"type":"journal_article","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>","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.","ista":"Henheik SJ, Lauritsen AB. 2025. Universal behavior of the BCS energy gap. Journal of Spectral Theory. 15(1), 305–352.","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>","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.","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>."},"department":[{"_id":"LaEr"},{"_id":"RoSe"}],"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","publication_identifier":{"eissn":["1664-0403"]},"doi":"10.4171/JST/540","day":"09","_id":"19548","issue":"1","file_date_updated":"2025-04-11T09:13:31Z","publication":"Journal of Spectral Theory","publisher":"EMS Press","date_published":"2025-01-09T00:00:00Z","scopus_import":"1","abstract":[{"lang":"eng","text":"We consider the BCS energy gap „.T / (essentially given by „.T / \u0019 .T; p\u0016/,\r\nthe BCS order parameter) at all temperatures 0 \u0014 T \u0014 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 ."}],"year":"2025","month":"01","isi":1,"arxiv":1,"date_updated":"2026-04-07T12:37:11Z","oa_version":"Published Version","file":[{"file_size":779158,"date_updated":"2025-04-11T09:13:31Z","success":1,"content_type":"application/pdf","file_name":"Henheik_JSpectralTheory_2025.pdf","checksum":"f49e06e8dba819f7ad52a202e287ebca","access_level":"open_access","creator":"cchlebak","date_created":"2025-04-11T09:13:31Z","relation":"main_file","file_id":"19549"}],"publication_status":"published","OA_place":"publisher","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020"},{"grant_number":"I06427","_id":"bda63fe5-d553-11ed-ba76-a16e3d2f256b","name":"Mathematical Challenges in BCS Theory of Superconductivity"}],"article_processing_charge":"No","external_id":{"arxiv":["2312.11310"],"isi":["001438931600009"]},"author":[{"id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","last_name":"Henheik","orcid":"0000-0003-1106-327X","first_name":"Sven Joscha","full_name":"Henheik, Sven Joscha"},{"id":"e1a2682f-dc8d-11ea-abe3-81da9ac728f1","orcid":"0000-0003-4476-2288","last_name":"Lauritsen","full_name":"Lauritsen, Asbjørn Bækgaard","first_name":"Asbjørn Bækgaard"}],"language":[{"iso":"eng"}],"intvolume":"        15","status":"public","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"DOAJ_listed":"1","page":"305–352","license":"https://creativecommons.org/licenses/by/4.0/","title":"Universal behavior of the BCS energy gap","quality_controlled":"1","has_accepted_license":"1","OA_type":"gold","related_material":{"record":[{"id":"19540","status":"public","relation":"dissertation_contains"}]},"volume":15,"ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_type":"original","corr_author":"1","ddc":["500"],"date_created":"2025-04-11T09:19:28Z"},{"_id":"14662","day":"25","publication":"Journal of Spectral Theory","file_date_updated":"2023-12-11T12:03:12Z","issue":"3","type":"journal_article","citation":{"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.","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>","short":"R. Seiringer, Journal of Spectral Theory 13 (2023) 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>.","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>"},"department":[{"_id":"RoSe"}],"oa":1,"doi":"10.4171/JST/469","publication_identifier":{"eissn":["1664-0403"],"issn":["1664-039X"]},"oa_version":"Published Version","file":[{"content_type":"application/pdf","file_name":"2023_JST_Seiringer.pdf","checksum":"9ce96ca87d56ea9a70d2eb9a32839f8d","file_size":201513,"success":1,"date_updated":"2023-12-11T12:03:12Z","file_id":"14677","access_level":"open_access","creator":"dernst","date_created":"2023-12-11T12:03:12Z","relation":"main_file"}],"article_processing_charge":"Yes","OA_place":"publisher","publication_status":"published","date_published":"2023-11-25T00:00:00Z","publisher":"EMS Press","arxiv":1,"date_updated":"2025-09-09T13:40:33Z","isi":1,"month":"11","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."}],"scopus_import":"1","year":"2023","page":"1045-1055","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"DOAJ_listed":"1","quality_controlled":"1","title":"Absence of excited eigenvalues for Fröhlich type polaron models at weak coupling","language":[{"iso":"eng"}],"external_id":{"isi":["001127467700008"],"arxiv":["2210.17123"]},"author":[{"orcid":"0000-0002-6781-0521","last_name":"Seiringer","full_name":"Seiringer, Robert","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87"}],"status":"public","intvolume":"        13","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2023-12-10T23:00:59Z","ddc":["510"],"corr_author":"1","article_type":"original","OA_type":"diamond","has_accepted_license":"1","volume":13},{"status":"public","intvolume":"        12","language":[{"iso":"eng"}],"external_id":{"arxiv":["2201.08090"],"isi":["000997933500008"]},"author":[{"first_name":"Christian","full_name":"Hainzl, Christian","last_name":"Hainzl"},{"id":"5DA90512-D80F-11E9-8994-2E2EE6697425","orcid":"0000-0002-9071-5880","last_name":"Roos","first_name":"Barbara","full_name":"Roos, Barbara"},{"id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","full_name":"Seiringer, Robert","first_name":"Robert","last_name":"Seiringer","orcid":"0000-0002-6781-0521"}],"quality_controlled":"1","title":"Boundary superconductivity in the BCS model","page":"1507–1540","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"ec_funded":1,"volume":12,"related_material":{"record":[{"id":"14374","relation":"dissertation_contains","status":"public"}]},"has_accepted_license":"1","date_created":"2023-07-10T16:35:45Z","ddc":["530"],"corr_author":"1","article_type":"original","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","doi":"10.4171/JST/439","publication_identifier":{"issn":["1664-039X"],"eissn":["1664-0403"]},"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.","citation":{"ista":"Hainzl C, Roos B, Seiringer R. 2023. Boundary superconductivity in the BCS model. Journal of Spectral Theory. 12(4), 1507–1540.","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>","short":"C. Hainzl, B. Roos, R. Seiringer, Journal of Spectral Theory 12 (2023) 1507–1540.","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>","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>.","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."},"department":[{"_id":"GradSch"},{"_id":"RoSe"}],"type":"journal_article","oa":1,"publication":"Journal of Spectral Theory","file_date_updated":"2023-07-11T08:19:15Z","issue":"4","_id":"13207","day":"18","date_updated":"2026-04-07T13:27:39Z","arxiv":1,"isi":1,"month":"05","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."}],"year":"2023","scopus_import":"1","date_published":"2023-05-18T00:00:00Z","publisher":"EMS Press","article_processing_charge":"No","project":[{"name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"694227"}],"publication_status":"published","oa_version":"Published Version","file":[{"file_id":"13208","relation":"main_file","date_created":"2023-07-11T08:19:15Z","creator":"alisjak","access_level":"open_access","checksum":"5501da33be010b5c81440438287584d5","file_name":"2023_EMS_Hainzl.pdf","content_type":"application/pdf","date_updated":"2023-07-11T08:19:15Z","success":1,"file_size":304619}]}]