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Which magnetic fields support a zero mode?

Frank, Rupert L. and Loss, Michael (2022) Which magnetic fields support a zero mode? Journal für die reine und angewandte Mathematik . ISSN 1435-5345. doi:10.1515/crelle-2022-0015. (In Press) https://resolver.caltech.edu/CaltechAUTHORS:20211004-222705713

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Abstract

This paper presents some results concerning the size of magnetic fields that support zero modes for the three-dimensional Dirac equation and related problems for spinor equations. It is a well-known fact that for the Schrödinger equation in three dimensions to have a negative energy bound state, the 3/2 norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3/2 norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1515/crelle-2022-0015DOIArticle
https://arxiv.org/abs/2012.13646arXivDiscussion Paper
ORCID:
AuthorORCID
Frank, Rupert L.0000-0001-7973-4688
Loss, Michael0000-0001-5008-3340
Additional Information:© 2022 Walter de Gruyter GmbH, Berlin/Boston. Received: 2021-01-29; Revised: 2021-11-29; Published Online: 2022-04-28. The authors would like to thank H. Kovarik and M. Lewin for helpful remarks. Partial support through U.S. National Science Foundation grants DMS-1363432 and DMS-1954995 (Rupert L. Frank) and DMS-1856645 (Michael Loss) and through the Deutsche Forschungsgemeinschaft (German Research Foundation) through Germany’s Excellence Strategy EXC-2111-390814868 (Rupert L. Frank) is acknowledged.
Funders:
Funding AgencyGrant Number
NSFDMS-1363432
NSFDMS-1954995
NSFDMS-1856645
Deutsche Forschungsgemeinschaft (DFG)EXC-2111-390814868
Classification Code:2010 Mathematics Subject Classification. Primary: 35F50; Secondary: 81V45, 47J10.
DOI:10.1515/crelle-2022-0015
Record Number:CaltechAUTHORS:20211004-222705713
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20211004-222705713
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111200
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:04 Oct 2021 22:40
Last Modified:17 May 2022 16:36

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