Equality of the spectral and dynamical definitions of reflection

Jonathan Breuer; Eric Ryckman; Barry Simon

doi:10.1007/s00220-009-0945-7

Equality of the spectral and dynamical definitions of reflection

Jonathan Breuer, Eric Ryckman, Barry Simon^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

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 = -∞ as t → -∞ goes entirely to x = ∞ as t → ∞. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.

Original language	American English
Pages (from-to)	531-550
Number of pages	20
Journal	Communications in Mathematical Physics
Volume	295
Issue number	2
DOIs	https://doi.org/10.1007/s00220-009-0945-7
State	Published - Feb 2010
Externally published	Yes

Bibliographical note

Funding Information:
Supported in part by NSF grant DMS-0652919.

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AB - 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 = -∞ as t → -∞ goes entirely to x = ∞ as t → ∞. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.

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Equality of the spectral and dynamical definitions of reflection

Abstract

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