TY - JOUR

T1 - Equality of the spectral and dynamical definitions of reflection

AU - Breuer, Jonathan

AU - Ryckman, Eric

AU - Simon, Barry

N1 - Funding Information:
Supported in part by NSF grant DMS-0652919.

PY - 2010/2

Y1 - 2010/2

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=77949656968&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0945-7

DO - 10.1007/s00220-009-0945-7

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AN - SCOPUS:77949656968

SN - 0010-3616

VL - 295

SP - 531

EP - 550

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -