A generalized Calderon Formula for open-arc diffraction problems: theoretical considerations

Abstract

We deal with the general problem of scattering by open-arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the form $\tilde{N} \tilde{S}[\varphi] = f$, where $\tilde{N}$ and $\tilde{S}$ are first-kind integral operators whose composition gives rise to a generalized Calder\'on formula of the form $\tilde{N} \tilde{S} = \tilde{J}_0^\tau + \tilde{K}$ in a {\em weighted, periodized} Sobolev space. The $\tilde{N} \tilde{S}$ formulation provides, for the first time, a second-kind integral equation for the open-arc scattering problem with Neumann boundary conditions. Numerical experiments show that, for both the Dirichlet and Neumann boundary conditions, our second-kind integral equations have spectra that are bounded away from zero and infinity as $k\to \infty$; to the authors' knowledge these are the first integral equations for these problems that possess this desirable property. Our proofs rely on three main elements: 1) Algebraic manipulations enabled by the presence of integral weights; 2) Use of the classical result of continuity of the Ces\`aro operator; and 3) Explicit characterization of the point spectrum of $\tilde{J}^\tau_0$, which, interestingly, can be decomposed into the union of a countable set and an open set, both tightly clustered around -1/4. As shown in a separate contribution, the new approach can be used to construct simple spectrally-accurate numerical solvers and, when used in conjunction with Krylov-subspace solvers such as GMRES, gives rise to dramatic reductions of Krylov-subspace iteration numbers vs. those required by other approaches.