Equality of the spectral and dynamical definitions of reflection

Jonathan Breuer, Eric Ryckman, Barry Simon*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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 languageAmerican English
Pages (from-to)531-550
Number of pages20
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Feb 2010
Externally publishedYes

Bibliographical note

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


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