Asymptotic and Spectral Analysis of Applied Non-self-adjoint Problems

该项目的目标是开发数学工具，用于研究物理学和现代数学生物学中出现的几个问题。前两个问题涉及波在非均匀介质和光纤中的传播。研究结果可用于隐藏目标的检测、无损检测以及噪声环境下信号传输质量的评估。此外，还将为生物制药学家用来描述活细胞过程的分子马达开发一个数学模型。该模型将证明该领域的某些假设是正确的，并将定量描述假设成立的参数。最后一个问题是对现代种群动力学模型的数学分析，旨在解释新物种的非本地和快速繁殖机制。在更技术性的层面上，该项目包括解决上述应用问题所需的均质化问题、非标准哈密顿算子的谱理论和奇异摄动偏微分方程。 具体而言，将获得以下结果。建立了（非自伴）内传播问题特征值计数函数的外尔定律。本文将证明具有随机扰动表面的波导的局部化定理。均匀化原理对于分子马达是合理的。将研究一般接触过程的波前传播。

The objective of this project is the development of mathematical tools for studying several problems emerging from physics and modern mathematical biology. The first two problems concern propagation of waves in non-homogeneous media and optical fibers. The results can be used for detection of hidden objects, nondestructive testing and estimation of the quality of signal transmission in the noisy environment. Further, a mathematical model will be developed for molecular motors used by biophysicists to describe processes in living cells. The model will justify certain hypotheses in the area and will describe quantitatively the parameters for which the hypotheses hold. The last problem concerns the mathematical analysis of modern models of population dynamics aiming to explain the mechanism for non-local and fast propagation of new species. The existence of stable populations with competition between species will be also explained.On a more technical level, the project includes problems in homogenization, spectral theory of non-standard Hamiltonians, and singularly perturbed partial differential equations needed to address the applied problems mentioned above. In particular, the following results will be obtained. The Weyl law for the counting function of eigenvalues of (non-self-adjoint) interior transmission problem will be established. Localization theorem will be proved for waveguides with randomly perturbed surfaces. The homogenization principle will be justified for molecular motors. Front propagation will be studied for general contact processes.