Geometric and obstacle scattering at low energy

Abstract

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case of functions eigenvalues generally exist at the bottom of the continuous spectrum and the corresponding eigenforms represent cohomology classes. We show that these eigenforms appear in the expansion of the resolvent, the scattering matrix, and the spectral measure in terms of the spectral parameter λ near zero, and we determine the first terms in this expansion explicitly. In dimension two an additional cohomology class appears as a resonant state in the presence of an obstacle. In even dimensions the expansion is in terms of λ and log λ. The theory of Hahn holomorphic functions is used to describe these expansions effectively. We also give a Birman-Krein formula in this context. The case of one forms with relative boundary conditions has direct applications in physics as it describes the scattering of electromagnetic waves.

The Classical and Quantum Photon Field for Non-compact Manifolds with Boundary and in Possibly Inhomogeneous Media

“…Elements in H 1 ( ) are harmonic one-forms satisfying relative boundary conditions. The result [51,Theorem 1.12] states that the image of H 0 (S d−1 ) in H 1 ( ) is spanned by a unique harmonic one form which I would like to describe more explicitly now in the most interesting case d = 3.…”

“…Then u =ũ −lim λ→0 ( +λ 2 ) −1 ũ is harmonic and solves the Dirichlet problem. It has the claimed decay at infinity because of the decay of the free Green's function as shown be the representation of the resolvent by gluing (see for example [51,Equ. (38)] or the limit absorption principle which is known to hold for this class of operators.…”

“…The Laplace operator =δ d + d δ defined with relative boundary conditions was analysed in the case = μ = 1 in [51] in great detail. I would like to note here that the analysis carried out in [51] carries over entirely to the more general case whenδ = τ −1 δ τ as long as τ − 1 is compactly supported. The reason is that only the structure near infinity and the form of the operator = (δ +d ) 2 was used.…”

“…as λ → 0 + , where and γ is the Euler-Mascheroni constant and β is a constant determined by the geometry and given in [51]. Here Q = 0 in case ∂O = ∅ or p = 1.…”

“…Since the continuous spectrum in this case is [0, ∞) there is no spectral gap and this makes it more difficult to modify the construction to project out possibly zero modes. In fact it was shown recently in [51] that the zero modes still appear in the expansions of generalised eigenfunctions that describe the continuous spectrum, illustrating that there is no clear cut between the continuous and the discrete spectrum (see also [29][30][31]). Whereas in the case when ( , h) is complete (which was essentially the case dealt with in [20,21]) one can still use the orthogonal projection onto the L 2 -kernel, this approach does not work if there is a boundary, i.e.…”