An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators

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.

21 pages

References in corpus (1)

• Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators

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• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• A spectral isoperimetric inequality for cones
• Eigenvalue curves for generalized MIT bag models
• Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions
• Limiting absorption principle and scattering matrix for Dirac operators with $\delta$-shell interactions
• Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors