Manifold decompositions and indices of Schr\"odinger operators
Manifold decompositions and indices of Schr\"odinger operators
复制标题
Schr"odinger算子的流形分解和指数
DOI：
10.1512/iumj.2017.66.6129
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Marzuola
中科院分区：
文献类型：
--
作者：
G. Cox;C. Jones;J. Marzuola
利用Maslov指标计算了紧致流形上Schr dinger算子不同边值问题的谱.主要结果是一个谱分解公式的流形$M$分为组件$\Omega_1 $和$\Omega_2 $的分离超曲面$\Sigma$。同伦变元将M上的二阶椭圆算子的谱与其在\Omega_1 $和\Omega_2 $上的狄利克雷谱和诺伊曼谱联系起来，差值由拉格朗日子空间的路径的马斯洛夫指数给出。这个Maslov指数可以用$\Sigma$上Dirichlet-to-Neumann映射的莫尔斯指数来表示。应用程序给出加倍结构，周期性边界条件和节点域的计数。特别地，给出了Courant节域定理的一个新的证明，并给出了节亏的一个显式公式。
The Maslov index is used to compute the spectra of different boundary value problems for Schr\"{o}dinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into components $\Omega_1$ and $\Omega_2$ by a separating hypersurface $\Sigma$. A homotopy argument relates the spectrum of a second-order elliptic operator on $M$ to its Dirichlet and Neumann spectra on $\Omega_1$ and $\Omega_2$, with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on $\Sigma$. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant's nodal domain theorem is given, with an explicit formula for the nodal deficiency.