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A Sharp Upper Bound on the Spectral Gap for Graphene Quantum Dots
Mathematical Physics, Analysis and Geometry ( IF 1 ) Pub Date : 2019-04-08 , DOI: 10.1007/s11040-019-9310-z
Vladimir Lotoreichik , Thomas Ourmières-Bonafos

The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected C3-domains with infinite mass boundary conditions. This bound is given in terms of a conformal variation, explicit geometric quantities and of the first eigenvalue for the disk. Its proof relies on the min-max principle applied to the squares of these Dirac operators. A suitable test function is constructed by means of a conformal map. This general upper bound involves the norm of the derivative of the underlying conformal map in the Hardy space ℋ2(D)$\mathcal {H}^{2}(\mathbb {D})$. Then, we apply known estimates of this norm for convex and for nearly circular, star-shaped domains in order to get explicit geometric upper bounds on the eigenvalue. These bounds can be re-interpreted as reverse Faber-Krahn-type inequalities under adequate geometric constraints.

中文翻译:

石墨烯量子点光谱间隙的尖锐上限

本文的主要结果是在具有无限质量边界条件的二维简单连接 C3 域中 Dirac 算子的第一个正特征值的尖锐上限。这个界限是根据共形变化、显式几何量和圆盘的第一个特征值给出的。它的证明依赖于应用于这些狄拉克算子平方的最小-最大原则。一个合适的测试函数是通过共形图构建的。这个一般上限涉及哈代空间 ℋ2(D)$\mathcal {H}^{2}(\mathbb {D})$ 中底层共形映射的导数的范数。然后,我们对凸域和近圆形的星形域应用该范数的已知估计,以获得特征值的明确几何上界。
更新日期:2019-04-08
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