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 \(\Omega \), 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 \(\epsilon \)-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.

我们获得了与拉普拉斯算子的第一个特征值的上限，该拉普拉斯算子与一个闭合的电势1型（因此具有零磁场）相关，作用于具有磁Neumann边界条件的平面域\（\ Omega \）的复函数。众所周知，只要电位允许至少一个非积分通量，则第一特征值是正的。通过量规不变性，如果简单地连接域，则最低特征值就简单地为零；反之，则为零。然后，我们仅根据孔数与面积之比得出基态能量的上限；对数值常数取模，其上限是尖锐的，并且我们表明，实际上，对以最大\（\ epsilon \）插入的域起作用的Aharonov-Bohm型算子可以实现相等（取常数）-网。在最后一部分中，我们表明可以完善上限，只要可以通过执行许多具有足够小的总长度的剪切操作，就可以在一个简单连接的域中变换给定的域；因此，我们得到最低特征值的上限，即孔数与面积之比乘以Cheeger型常数，该上限在域在度量上接近简单连接的域时趋于零。