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Which magnetic fields support a zero mode?

Frank, Rupert L. and Loss, Michael (2020) Which magnetic fields support a zero mode? . (Unpublished)

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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 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:Report or Paper (Discussion Paper)
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Frank, Rupert L.0000-0001-7973-4688
Additional Information:© 2020 by the authors. Reproduction of this article by any means permitted for non-commercial purposes. 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 (R.L.F.) is acknowledged as is partial support through grant DMS-1856645 (M.L.).
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Classification Code:2010 Mathematics Subject Classification. Primary: 35F50; Secondary: 81V45, 47J10.
Record Number:CaltechAUTHORS:20211004-222705713
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111200
Deposited By: George Porter
Deposited On:04 Oct 2021 22:40
Last Modified:04 Oct 2021 22:40

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