Translation Representations and Scattering By Two Intervals

TL;DR

This paper develops a model for obstacle scattering using translation representations in the complement of two intervals, providing spectral analysis and scattering matrices for systems in wave and quantum contexts.

Contribution

It introduces a novel approach to obstacle scattering modeling via translation representations for two-interval complements, with detailed spectral and scattering analysis.

Findings

Constructed a unitary model for obstacle scattering.

Derived spectral representations and scattering matrices.

Applied results to acoustic wave and quantum tunneling models.

Abstract

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.