Lattice Dislocations in a 1-Dimensional Model

Abstract:The spectral properties of the Schrödinger operator T(t)=−d2/dx2+q(x,t) in L2(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 σac(T(t))=σac(T(0)) consists of intervals, which are separated by the gaps γn(T(t))=γn(T(0))=(αn−,αn+), n≥1. We prove: in each gap γn≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λn±(t) of the dislocation operator, such that λn±(0)=αn± and the point λn±(t) runs clockwise around the gap γn changing the energy sheet whenever it hits αn±, making n/2 complete revolutions in unit time. On the first sheet λn±(t) is an eigenvalue and on the second sheet λn±(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 λn±(t) as n→∞ is determined.