Equality of the Spectral and Dynamical Definitions of Reflection

For full-line Jacobi matrices, Schrödinger 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.

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