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

这个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。其结果可用于检测隐藏目标、无损检测和评估噪声环境中的信号传输质量。此外，将为生物物理学家用来描述活细胞中的过程的分子马达开发一个数学模型。该模型将证明该地区的某些假设是正确的，并将定量地描述这些假设所适用的参数。最后一个问题涉及现代种群动力学模型的数学分析，旨在解释新物种非本地快速繁殖的机制。在更技术性的层面上，该项目包括齐次化问题、非标准哈密顿量的谱理论以及解决上述应用问题所需的奇异摄动偏微分方程组。特别是，将获得以下结果。建立(非自伴)内传输问题本征值计数函数的Weyl定律。证明了具有随机扰动表面的波导的局部化定理。对于分子马达，均一化原理是合理的。我们将研究一般接触过程中的前沿传播。

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.