论文标题
弗雷德·霍尔姆（Fredholm
Fredholm determinants, Evans functions and Maslov indices for partial differential equations
论文作者
论文摘要
Evans函数是一个众所周知的工具，用于在一个空间维度中定位差分运算符的光谱。在本文中，我们构建了一个多维类似物，作为边界上迪里奇特与罗宾蛋白算子的比率的修改后的决定因素。这提供了一个工具，用于研究不需要是自动化的二阶椭圆运算符的特征值计数功能。为此，我们将局部表示理论用于近形算子值铅笔，并将椭圆形操作员的特征值的代数多重性与抢劫 - 抢劫和抢劫 - 罗宾与robin-to-dirichlet operator铅笔相关联。在自动化案例中，我们将构造与马斯洛夫指数（Maslov Index）联系起来，Maslov指数是差异操作员光谱理论中的另一个知名工具。这为Maslov指数提供了新的见解，并使我们能够通过复杂的分析方法获得关键的单调性结果。
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.
