{"degree_awarded":"PhD","status":"public","_id":"14374","file":[{"relation":"main_file","file_id":"14398","content_type":"application/pdf","checksum":"ef039ffc3de2cb8dee5b14110938e9b6","file_name":"phd-thesis-draft_pdfa_acrobat.pdf","date_created":"2023-10-06T11:35:56Z","file_size":2365702,"creator":"broos","access_level":"open_access","date_updated":"2023-10-06T11:35:56Z"},{"date_created":"2023-10-06T11:38:01Z","date_updated":"2023-10-06T11:38:01Z","access_level":"closed","creator":"broos","file_size":4691734,"content_type":"application/x-zip-compressed","file_id":"14399","relation":"source_file","file_name":"Version5.zip","checksum":"81dcac33daeefaf0111db52f41bb1fd0"}],"file_date_updated":"2023-10-06T11:38:01Z","ec_funded":1,"doi":"10.15479/at:ista:14374","oa_version":"Published Version","department":[{"_id":"GradSch"},{"_id":"RoSe"}],"project":[{"name":"Analysis of quantum many-body systems","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","grant_number":"694227","call_identifier":"H2020"},{"grant_number":"I06427","_id":"bda63fe5-d553-11ed-ba76-a16e3d2f256b","name":"Mathematical Challenges in BCS Theory of Superconductivity"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","type":"dissertation","date_updated":"2023-10-27T10:37:30Z","month":"09","date_created":"2023-09-28T14:23:04Z","publication_status":"published","alternative_title":["ISTA Thesis"],"publisher":"Institute of Science and Technology Austria","license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","supervisor":[{"id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","full_name":"Seiringer, Robert","last_name":"Seiringer","orcid":"0000-0002-6781-0521","first_name":"Robert"}],"date_published":"2023-09-30T00:00:00Z","article_processing_charge":"No","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"13207"},{"id":"10850","relation":"part_of_dissertation","status":"public"}]},"title":"Boundary superconductivity in BCS theory","citation":{"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.","ama":"Roos B. Boundary superconductivity in BCS theory. 2023. doi: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.","apa":"Roos, B. (2023). Boundary superconductivity in BCS theory. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:14374","ieee":"B. Roos, “Boundary superconductivity in BCS theory,” Institute of Science and Technology Austria, 2023."},"ddc":["515","539"],"publication_identifier":{"issn":["2663 - 337X"]},"has_accepted_license":"1","tmp":{"short":"CC BY-NC-SA (4.0)","image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode"},"day":"30","language":[{"iso":"eng"}],"author":[{"first_name":"Barbara","orcid":"0000-0002-9071-5880","last_name":"Roos","full_name":"Roos, Barbara","id":"5DA90512-D80F-11E9-8994-2E2EE6697425"}],"page":"206","oa":1,"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."}],"year":"2023"}