标题： 狄利克雷到诺伊曼算子在二球面上的谱聚类渐近行为 显示英文标题

标题： Spectral cluster asymptotics of the Dirichlet to Neumann operator on the two-sphere

摘要： 我们研究了与单位球中的Schrödinger算子相关的二维球面的Dirichlet到Neumann算子的谱。 谱在自然数序列$k=1,2,\ldots$周围形成大小为$O(1/k)$的簇， 并且我们计算了在$k\to\infty$时簇内特征值渐近分布的前三项（带不变量）。 证明有两个独立方面。 第一个是对应用平均方法后出现的Dirichlet到Neumann算子的Berezin符号的研究。 第二个是使用球面上闭测地线流形上的Berezin-Toeplitz算子的符号演算。 显示英文摘要

摘要： We study the spectrum of the Dirichlet to Neumann operator of the two-sphere associated to a Schr\"odinger operator in the unit ball. The spectrum forms clusters of size $O(1/k)$ around the sequence of natural numbers $k=1,2,\ldots$, and we compute the first three terms in the asymptotic distribution of the eigenvalues within the clusters, as $k\to\infty$ (band invariants). There are two independent aspects of the proof. The first is a study of the Berezin symbol of the Dirichlet to Neumann operator, which arises after one applies the averaging method. The second is the use of a symbolic calculus of Berezin-Toeplitz operators on the manifold of closed geodesics of the sphere.