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arXiv 1111.4549math-phmath.MP
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磁性量子点中二维无质量狄拉克费米子的局域化

Localization of two-dimensional massless Dirac fermions in a magnetic quantum dot

Martin Könenberg, Edgardo Stockmeyer

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摘要：

考虑二维无质量Dirac算子$H$在扰动的均匀磁场$B=B_0+b$和电势$V$存在下的解.对于$V\in L_{\rm loc}^p(\R^2)$，$p\in(2,\infty]$，和$b\in L_{\rm loc}^q(\R^2)$，$q\in(1,\infty]$，都衰减到无穷远，我们表明，在离散谱的$H$的状态是超指数本地化。假设V=0，我们建立了在第零和第一朗道能级之间存在这样的状态。另外，在$b$是旋转对称的，且$V$满足一定的角变量解析性条件的条件下，我们证明了属于$H$离散谱的态是类高斯局域态.

英文摘要：

We consider a two-dimensional massless Dirac operator $H$ in the presence of a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric potential $V$. For $V\in L_{\rm loc}^p(\R^2)$, $p\in(2,\infty]$, and $b\in L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that states in the discrete spectrum of $H$ are superexponentially localized. We establish the existence of such states between the zeroth and the first Landau level assuming that V=0. In addition, under the condition that $b$ is rotationally symmetric and that $V$ satisfies certain analyticity condition on the angular variable, we show that states belonging to the discrete spectrum of $H$ are Gaussian-like localized.

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