Abstract:
The spectral properties of the Schrödinger operator T(t)=−d 2/dx 2+q(x,t) in L 2(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ ac (T(t))=σ ac (T(0)) consists of intervals, which are separated by the gaps γ n (T(t))=γ n (T(0))=(α n −,α n +), n≥1. We prove: in each gap γ n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ n ±(t) of the dislocation operator, such that λ n ±(0)=α n ± and the point λ n ±(t) runs clockwise around the gap γ n changing the energy sheet whenever it hits α n ±, making n/2 complete revolutions in unit time. On the first sheet λ n ±(t) is an eigenvalue and on the second sheet λ n ±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ0(t) in the basic gap γ0(T(t))=γ0(T(0))=(−∞ ,α0 +). The asymptotics of λ n ±(t) as n→∞ is determined.
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Received: 5 April 1999 / Accepted: 3 March 2000
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Korotyaev, E. Lattice Dislocations in a 1-Dimensional Model. Commun. Math. Phys. 213, 471–489 (2000). https://doi.org/10.1007/PL00005529
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DOI: https://doi.org/10.1007/PL00005529