Maslov指数与Monodromy

Fredholm determinants, Evans functions and Maslov indices for partial differential equations

The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin…

一维$θ-$周期薛定谔算子的Maslov指数计数谱

Counting spectrum via the Maslov index for one dimensional $θ-$periodic Schrödinger operators

研究了有限区间上具有$n\times n$矩阵值位势的Schr dinger算子在$θ-$周期边界条件下的谱.对于两个这样的算子，对应于不同的$θ$值，我们计算其特征值计数函数的差通过路径的拉格朗日平面的Maslov指数。此外，我们还导出了本征值关于$θ$的Maslov交叉形式的导数公式。最后，我们给出了一个新的较短的证明最近的结果有关的Morse和Maslov指数的薛定谔算子为固定的$θ$。