论文标题
平面域上具有零磁场的Laplacian基态能量的上限
Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains
论文作者
论文摘要
我们获得了与封闭电位$ 1 $ - 形式（因此，零磁场）相关的磁性拉普拉斯的第一个特征值的上限，作用于平面域$ω$的复杂函数，具有磁性Neumann边界条件。众所周知，每当潜力至少接收到一个非整合通量时，第一个特征值是正的。通过规格不变性，如果域仅连接，则最低特征值只是零。然后，我们仅取决于孔数和面积之间的比率，获得基态能量的上限； Modulo一个数值常数上限很尖锐，我们表明，对于Aharonov-bohm型运算符的aharonov-bohm型操作员实际上是平等性的（Modulo a是常数）。在最后一部分中，我们表明可以改进上限，前提是一个人可以通过执行足够小的总长度进行多个切割来转换给定域。因此，我们通过孔数和面积之间的比率获得了最低特征值的上限，乘以cheeger型常数，当域在域上靠近简单连接的域时，该常数往往为零。
We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential $1$-form (hence, with zero magnetic field) acting on complex functions of a planar domain $Ω$, with magnetic Neumann boundary conditions. It is well-known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $ε$-net. In the last part we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.
