Spectral Problems on Families of Domains and Operator M-functions

在磁流体力学、量子力学、量子图论和应用数学的许多领域中，控制方程是所谓的椭圆偏微分方程。这些方程在由物理边界所划分的某些区域（称为域）中有效，在该区域上必须满足某些边界条件。有时域是外部的：它是围绕某些障碍物的区域。在其他情况下，边界上尖点和角点的存在意味着偏微分方程的解在尖点或角点附近可能表现出“不良行为”。然而，在远离边界的地方，解表现良好，我们可以想象它们在域内画的虚边界上满足很好的规则边界条件。那么我们如何描述所有的边界条件我们必须想象强加在这些想象的边界上，以捕捉在真实物理边界附近偏微分方程解的所有可能的奇怪行为？我们该怎么处理这些结果呢？第一个问题要求我们发展一个边界值空间的抽象理论。第二，我们想要发展一个狄利克雷到诺伊曼映射的抽象理论，或者m算子：这是一个映射，当我们知道它的值时，它告诉我们解的梯度。我们想了解这个图是如何依赖于方程中的物理参数的。这些参数中的一些被称为本征参数，这些参数的临界值，被称为本征值，它描述，例如，系统的自然共振频率，或者它从稳定到不稳定的能量。我们想要理解当我们将虚（外）边界移动到实（非光滑）边界组件或无穷大时，边界的一些不变（内）组件（例如光滑障碍物）上的m算子是如何变化的；以及它对系统特征值的影响。最重要的是，我们要对所谓的非自伴随问题做所有这些，其中特征值可能是复数。

In magnetohydrodynamics, in quantum mechanics, in quantum graph theory and in many areas of applied mathematics, the governing equations are so-called elliptic PDEs. These equations are to be valid in some region - called a domain - delimited by a physical boundary, upon which certain boundary conditions must be satisfied. Sometimes the domain is exterior: it is the region surrounding some obstacle. In other cases the presence of cusps and corners on the boundary means that solutions of the PDEs may exhibit `bad behaviour' near the cusps or corners. However, away from the boundary, the solutions are well behaved, and we can imagine that they would satisfy nice regular boundary conditions on an imaginary boundary drawn inside the domain. So how can we describe all the boundary conditions we would have to imagine imposing on these imaginary boundaries to capture all of the possible weird behaviours of the solutions of the PDEs near the real, physical boundary? And what would we do with the results? The first of these questions requires us to develop an abstract theory of boundary value spaces. For the second, we want to develop an abstract theory of a Dirichlet to Neumann map, or M-operator: this is the map which tells us the gradient of the solution whenever we know its values. We want to understand how this map might depend on physical parameters in the equations. Some of these parameters are called eigenparameters and there are critical values of these parameters, called eigenvalues, which describe, e.g., the natural resonant frequencies of the system, or the energies at which it passes from stable to unstable. We want to understand how the M-operators on some unchanging (inner) component of the boundary (say, a smooth obstacle) change as we move the imaginary (outer) boundary towards the real (non-smooth) boundary component, or to infinity; and the effect which this has on the eigenvalues of the system. Most importantly, we want to do all of this for so-called non-selfadjoint problems, where the eigenvalues may be complex.