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Published October 4, 2021 | Submitted
Journal Article Open

Which magnetic fields support a zero mode?


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.

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.

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