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型偏微分方程组。这些方程在由物理边界界定的某个区域--称为区域--是有效的，在该区域上必须满足某些边界条件。有时，领域是外部的：它是围绕着某个障碍的区域。在其他情况下，边界上尖角的存在意味着偏微分方程组的解可能在尖角附近表现出“不良行为”。然而，在边界之外，解表现得很好，我们可以想象它们将在区域内绘制的虚边界上满足良好的正则边界条件。那么，我们如何描述所有的边界条件，我们将不得不想象施加在这些想象的边界上，以捕捉在真实的物理边界附近偏微分方程组的解的所有可能的奇怪行为？我们将如何处理这些结果呢？这些问题中的第一个要求我们发展一个关于边值空间的抽象理论。对于第二个，我们想要发展一个抽象的Dirichlet到Neumann映射，或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。我们想要了解这张图可能如何依赖于方程中的物理参数。这些参数中的一些被称为本征参数，并且这些参数有临界值，称为本征值，它描述了例如系统的固有共振频率，或者从稳定到不稳定的能量。我们想要了解当我们将虚(外)边界移向实(非光滑)边界分量或向无穷远移动时，边界(比方说，光滑障碍)某个不变(内)分量上的M-算子如何变化；以及这对系统本征值的影响。最重要的是，我们想要为所谓的非自伴问题做所有这些，在这些问题中，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.