论文标题
平面域中磁场为零的Laplacian的第一个特征值的下限
Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains
论文作者
论文摘要
我们研究具有零磁场的Laplacian，其作用于平面域$ω$的复杂函数，并具有磁性Neumann边界条件。如果简单地连接$ω$，那么频谱将减少到通常的Neumann Laplacian的光谱；因此，我们专注于由凸曲线界定的乘积连接的域，并根据$ω$的几何形状和拓扑结束，证明其基态下限。除该区域，周长和直径外，在估计值中起着至关重要的作用的几何不变体是内孔周围的电势一体形的通量以及域的边界组件之间的距离。更确切地说，其最小宽度和最大宽度之间的比率。然后，我们给出了双连接域的下限，该域在该比率方面很清晰，并为具有任意孔数的域的一般下限。当内孔缩小到点时，我们作为推色获得了带有任意数量的杆的所谓的Aharonov-Bohm操作员的第一个特征值的下限。
We study the Laplacian with zero magnetic field acting on complex functions of a planar domain $Ω$, with magnetic Neumann boundary conditions. If $Ω$ is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of $Ω$. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles.
