Equality of the spectral and dynamical definitions of reflection. (English) Zbl 1190.47032
Summary: 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 \(x = - \infty\) as \(t \rightarrow - \infty\) goes entirely to \(x = \infty \) as \(t \rightarrow \infty\). This allows us to settle a conjecture of P.Deift and B.Simon [Commun.Math.Phys.90, 389–411 (1983; Zbl 0562.35026)] regarding ergodic Jacobi matrices.
MSC:
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 47A40 | Scattering theory of linear operators |
Keywords:
whole-line Jacobi matrices; two-sided ergodic Jacobi matrices; 1D Schrödinger operators; two-sided CMV matrices; spectrally reflectionless; dynamically reflectionlessCitations:
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