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 -