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，控制方程都是所谓的椭圆偏微分方程。这些方程在某个区域（称为区域）中是有效的，该区域由物理边界限定，在该物理边界上必须满足某些边界条件。有时候，这个领域是外部的：它是围绕着某个障碍物的区域。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可能在尖点或角点附近表现出“不良行为”。然而，远离边界，解表现良好，我们可以想象，它们将满足在域内绘制的假想边界上的规则边界条件。那么，我们如何描述所有的边界条件，我们必须想象，强加在这些假想的边界上，以捕捉靠近真实的物理边界的偏微分方程解的所有可能的奇怪行为？我们要拿结果怎么办？这些问题中的第一个要求我们发展边值空间的抽象理论。对于第二个，我们想发展一个狄利克雷到诺依曼映射的抽象理论，或M-算子：这是一个映射，它告诉我们解的梯度，只要我们知道它的值。我们想知道这个映射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特征参数，并且存在这些参数的临界值，称为特征值，其描述例如，系统的自然共振频率，或系统从稳定状态转变为不稳定状态时的能量。我们想了解当我们把虚（外）边界移向真实的（非光滑的）边界分量或无穷远时，边界的某个不变（内）分量（比如光滑障碍物）上的M算子如何变化;以及这对系统本征值的影响。最重要的是，我们想对所谓的非自伴问题做所有这些，其中特征值可能是复杂的。

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.