Fredholm determinants, Evans functions and Maslov indices for partial differential equations
Fredholm determinants, Evans functions and Maslov indices for partial differential equations
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DOI：
10.1007/s00208-023-02696-6
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发表时间：
2022-06
影响因子：
1.4
通讯作者：
G. Cox;Y. Latushkin;A. Sukhtayev
中科院分区：
文献类型：
--
作者：
G. Cox;Y. Latushkin;A. Sukhtayev
埃文斯函数是一个众所周知的工具，用于定位在一个空间维度的微分算子的谱。在本文中，我们构造了一个多维模拟作为修改的Fredholm行列式的比Dirichlet-to-Robin算子的边界上。这为研究不需要自伴的二阶椭圆算子的特征值计数函数提供了一个工具。为了做到这一点，我们使用亚纯算子值束的局部表示理论，并将椭圆算子的本征值的代数重数与Robin-to-Robin和Robin-to-Dirichlet算子束的代数重数联系起来。在自伴的情况下，我们将我们的建设与Maslov指数，另一个著名的工具，在谱理论的微分算子。这给了新的见解马斯洛夫指数，并允许我们获得重要的单调性结果，复杂的分析方法。
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-to-Dirichlet operator pencils. In the self-adjoint case we relate our construction to the Maslov index, another well known tool in the spectral theory of differential operators. This gives new insight into the Maslov index and allows us to obtain crucial monotonicity results by complex analytic methods.