Studying unitary one-parameter groups in Hilbert space (U(t), H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U(t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator and scattering matrix. We illustrate our results in the Lax-Phillips model where (U(t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states.
& Tian, F.
(2012). Translation Representations and Scattering by Two Intervals. Journal of Mathematical Physics, 53 (5), 53505.