Equality of the Spectral and Dynamical Definitions of Reflection

Abstract

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.

References in corpus (2)

• The absolutely continuous spectrum of one-dimensional Schr"odinger operators
• Right limits and reflectionless measures for CMV matrices

Cited by in corpus (10)

• A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula
• Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas
• Critical Lieb-Thirring Bounds in Gaps and the Generalized Nevai Conjecture for Finite Gap Jacobi Matrices
• Generalized reflection coefficients
• Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
• Right limits and reflectionless measures for CMV matrices
• A note on reflectionless Jacobi matrices
• Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions
• Reflectionless discrete Schrödinger operators are spectrally atypical
• Reflectionless CMV matrices and scattering theory