Spectral Problems on Families of Domains and Operator M-functions

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.