Spectral theory of differential and weighted composition operators

该项目的主要理论内容是研究具有无界算子系数的Fredholm微分算子及其Fredholm指标，并利用Banach空间上线性微分方程的动力学性质。这包括埃文斯函数作为Fredholm行列式的算子理论处理。在Banach空间值函数的Sobolev空间和Lebesgue空间上，将发展加权复合（发展）半群和相关抽象微分算子的谱分析的新方法。此外，我们还将研究加权复合算子和相关微分算子之间的联系，以及诸如演化问题解的最大正则性，Atiyah-Patodi-Singer谱流和指标定理，无限维莫尔斯理论，Krein-Birman谱移位函数和半可分核算子的Fredholm行列式等主题。该项目的主要应用部分是利用这些方法研究二维和三维经典线性化Euler算子的点谱和本质谱，该算子是通过对流体力学的Euler方程关于定常状态进行线性化而得到的，该项目处于复杂动力系统理论、多参数方程的数学理论、描述流体在平衡状态下行为的流体力学稳定性理论。我们将发展新的方法研究加权合成半群，数学模型描述各种影响的运输沿着轨迹的给定复杂系统。这些方法将被应用于欧拉方程的线性近似的研究，该方程在二维和三维中控制理想不可压缩流体的运动，这将增加我们对平衡附近的流体动力学的理解。

The main theoretical component of this project is to study Fredholm differential operators with unbounded operator coefficients and their Fredholm index in terms of dynamical properties of linear differential equations on Banach spaces. This includes an operator-theoretical treatment of Evans function as a Fredholm determinant. New methods will be developed in spectral analysis of weighted composition (evolution) semigroups and related abstract differential operators on Sobolev and Lebesgue spaces of Banach space-valued functions. Also, we will study connections between the weighted composition operators and associated differential operators, and topics such as maximal regularity of solutions of evolution problems, Atiyah-Patodi-Singer Spectral Flow and Index Theorem, infinite dimensional Morse Theory, Krein-Birman spectral shift function, and Fredholm determinants for operators with semi-separable kernels. The main applied component of this project is to use these methods to study the point and essential spectrum of the classical linearized Euler operator in dimensions two and three obtained by linearizing the Euler equations of hydrodynamics about a steady state.This project lies at the intersection of the theory of complex dynamical systems, the mathematical theory of equations containing many parameters, and stability theory of hydrodynamics describing the behavior of a fluid in equilibria. We will develop new methods of study of the weighted composition semigroups, the mathematical models describing various effects of transport along trajectories of the given complex system. These methods will be applied to the study of a linear approximation of the Euler equations governing the motion of an ideal incompressible fluid in dimensions two and three, which will add to our understanding of fluid dynamics near equilibria.