A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint

Abstract

We consider resolvents of operators taking the form A = Γ1BΓ1 where Γ1(k) is a projection that acts locally in Fourier space and B(x) is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites. We review how Q * -convex operators can be used to bound the spectrum of A. Then, based on the Cherkaev-Gibiansky transformation and subsequent developments, that we reformulate, we obtain for non-Hermitian B, a Stieltjes type integral representation for the resolvent (z0I − A) −1 . The representation holds in the half plane Re(e iϑ z0) > c, where ϑ and c are such that c − [e iϑ B + e −iϑ B † ] is positive definite (and coercive).

Some open problems in the theory of composites

“…Scattering problems can also be expressed in this form [73] by incorporating the fields "at infinity" appropriately. The analog for quadratic forms of quasiconvexity is then Q * -convexity: a quadratic form f (P) is Q * -convex if f (E) ≥ 0 for all E ∈ E. Q * -convex functions allow one to place bounds on the spectrum of the operator relevant to the problem [75,80]. The subject of Q * -convexity remains to be explored, and simple examples of Q * -convex functions need to be found for the various equations, beyond quasiconvex functions and those discovered for the Schrödinger equation (Sections 13.6 and 13.7 of [91]) .…”