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Equality of the Spectral and Dynamical Definitions of Reflection

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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.

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Authors and Affiliations

  1. Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91125, USA

    Jonathan Breuer, Eric Ryckman & Barry Simon

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  1. Jonathan Breuer
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  2. Eric Ryckman
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  3. Barry Simon
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Correspondence to Barry Simon.

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Communicated by M. Aizenman

© 2009 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Supported in part by NSF grant DMS-0652919.

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Breuer, J., Ryckman, E. & Simon, B. Equality of the Spectral and Dynamical Definitions of Reflection. Commun. Math. Phys. 295, 531–550 (2010). https://doi.org/10.1007/s00220-009-0945-7

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