Equality of the Spectral and Dynamical Definitions of Reflection

TL;DR

This paper proves that for certain operators, being reflectionless in terms of m-functions is equivalent to having no reflection in the dynamics, resolving a longstanding conjecture for ergodic Jacobi matrices.

Contribution

It establishes the equivalence between spectral and dynamical reflectionless properties for full-line operators, settling a 1983 conjecture.

Findings

Reflectionless property equals no dynamical reflection for these operators.

Resolved Deift and Simon's 1983 conjecture.

Unified spectral and dynamical perspectives on reflection.

Abstract

For full-line Jacobi matrices, Schr\"odinger operators, and CMV matrices, we show that being reflectionless, in the sense of the well-known property of $m$-functions, is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to $x=-\infty$ as $t\to -\infty $ goes entirely to $x=\infty$ as $t\to\infty$. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.