Research in operator theory and applied dynamical systems

这个项目的主要目的是深入研究埃文斯函数的算子论性质，埃文斯函数是当代行波和波型稳定性理论的主要工具之一。我们计划在无限维环境下发展一种关于抽象方程的Evans函数的形式，这类方程是对角一致稳定的双半群的有界扰动，包括(在应用方面)无限圆柱中的线性化抛物问题，以及许多其他问题。我们将利用Krein的谱移函数推导出抽象微分算子的Fredholm型指标的一个新公式。我们将探索Evans函数与Bman-Schwinger型积分算子的修正Fredholm型行列式之间的强大而令人惊讶的联系。这包括修正的Fredholm型行列式在特征值处导数的新抽象公式，以及用于检测沿脉冲线性化的不稳定特征值的Jost和Evans函数的导数的具体实现。特别是，我们将对一维固体燃烧波理论中的一个重要应用问题进行Evans函数谱分析。这项建议的主题是复杂动力系统应用理论和算子谱理论的相互作用，算符谱理论是包含无限多参数的方程的数学理论。我们将研究行波、脉冲、锋面和其他可能描述燃烧、波传播和许多其他自然现象的模式。我们的主要应用目标是开发工具来了解模式是否稳定，即它们的结构是否在小扰动下保持不变。将量子力学的一些思想引入这一领域，我们将把Evans函数的概念推广到包含无限多参数的系统中，Evans函数是一个类似于微分方程组理论中的Wronskian的行列式。我们的主要理论贡献将是证明统一定理，这些定理使用Evans函数，描述通过包含无限多参数的微分方程的渐近性质线性化的关于上述模式的方程的可解性，并与模式的稳定性有关。

The main objective of this project is to give an in-depth study of the operator-theoretic nature of the Evans function, one of the principal tools in contemporary stability theory of travelling waves and wave patterns. We plan to develop a formalism for the Evans function in the infinite dimensional setting for an abstract class of equations obtained as bounded perturbations of diagonal uniformly stable bi-semigroups, including (on the applied side) linearized parabolic problems in infinite cylinders, and many others. We will derive a new formula for the Fredholm index of abstract differential operators in terms of Krein's spectral shift function. We will explore powerful and surprising connections between Evans functions and the modified Fredholm determinants of the Birman-Schwinger-type integral operators. This includes a new abstract formula for the derivative of the modified Fredholm determinant at an eigenvalue, and its concrete realization for the derivatives of the Jost and Evans functions used to detect unstable eigenvalues of linearizations along pulses. In particular, we will carry out the Evans function spectral analysis of an important applied problem in the theory of combustion waves in one-dimensional solids.The topic of this proposal is an interplay of applied theory of complex dynamical systems and operator spectral theory, the mathematical theory of equations containing infinitely many parameters. We will study travelling waves, pulses, fronts, and other patterns potentially describing combustion, wave propagation, and many other natural phenomena. Our main applied goal is to develop tools for understanding whether the patterns are stable, that is, whether their structure is being preserved under small perturbations. Bringing in this area some ideas from quantum mechanics, we will generalize and utilize for systems containing infinitely many parameters the concept of the Evans function, a determinant similar to the Wronskian in the theory of differential equations. Our main theoretical contribution will be in the proof of unifying theorems that use the Evans function and describe solvability properties of equations linearized about the above-mentioned patterns via asymptotic properties of differential equations containing infinitely many parameters, and related to stability of the patterns.