We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a "rough" surface. The low-frequency robustness of the potential formulation of Maxwell's equations makes this model a promising starting point for Galerkin discretization.

We thoroughly analyse the double-layer potential's role in approaches to spectral sets in the spirit of Delyon-Delyon, Crouzeix and Crouzeix-Palencia. While the potential is well-studied, we aim to clarify on several of its aspects in light of these references. In particular, we illustrate how the associated integral operators can be used to characterize the convexity of the domain and the inclusion of the numerical range in its closure. We furthermore give a direct proof of a result by Putinar-Sandberg-a generalization of Berger-Stampfli's mapping theorem-circumventing dilation theory. Finally, we show for matrices that any smooth domain whose closure contains the numerical range admits a spectral constant only depending on the extremal function and vector. This constant is consistent with the so far best known absolute bound .

In this note the two dimensional Dirac operator with an electrostatic -shell interaction of strength supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths the continuous spectrum of inside the spectral gap of the free Dirac operator collapses abruptly to a single point.

In this paper we shall use realization theory, a favourite technique of Rien Kaashoek, to prove new results about a class of holomorphic functions on an annulus where . The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator to a normal operator with spectrum in . Their work suggested the following norm on the space of holomorphic functions on , By analogy with the classical Schur class of holomorphic functions with supremum norm at most 1 on the disc , it is natural to consider the of holomorphic functions of dp-norm at most 1 on . Our central result is a Pick interpolation theorem for functions in that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple of distinct interpolation nodes in , we introduce a special set of positive definite matrices, which we call . The DP Pick problem , is shown to be solvable if and only if, We prove further that a solvable DP Pick problem has a solution which is a rational function with a finite-dimensional model, an intriguing result which opens up the possibility of a theory of extremal functions from analogous to the theory of finite Blaschke products.