Abstract
In this article we investigate spectral properties of the coupling \({H + V_\lambda}\), where \({H = -i\alpha \cdot \nabla+m\beta}\) is the free Dirac operator in \({\mathbb{R}^3}\), \({m > 0}\) and \({V_\lambda}\) is an electrostatic shell potential (which depends on a parameter \({\lambda \in \mathbb{R}}\)) located on the boundary of a smooth domain in \({\mathbb{R}^3}\). Our main result is an isoperimetric-type inequality for the admissible range of \({\lambda}\)’s for which the coupling \({H + V_\lambda}\) generates pure point spectrum in \({(-m, m)}\). That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman–Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible \({\lambda}\)’s, and we use this to relate the endpoints of the admissible range of \({\lambda}\)’s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.
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Communicated by W. Schlag
Arrizabalaga was supported in part by MTM2011-24054 and IT641-13. Mas was supported by the Juan de la Cierva program JCI2012-14073 (MEC, Gobierno de España), ERC Grant 320501 of the European Research Council (FP7/2007-2013), MTM2011-27739 and MTM2010-16232 (MICINN, Gobierno de España), and IT-641-13 (DEUI, Gobierno Vasco). Vega was partially supported by SEV-2013-0323, MTM2011-24054 and IT641-13.
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Arrizabalaga, N., Mas, A. & Vega, L. An Isoperimetric-Type Inequality for Electrostatic Shell Interactions for Dirac Operators. Commun. Math. Phys. 344, 483–505 (2016). https://doi.org/10.1007/s00220-015-2481-y
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DOI: https://doi.org/10.1007/s00220-015-2481-y