Pavel Exner
The Absence of the Absolutely Continuous Spectrum for $\delta'$ Wannier-Stark Ladders
The paper is published: J. Math. Phys. 36, 9 (1995) 4561-4570

MSC:

34L40 Particular operators (Dirac, one-dimensional Schrodinger, etc.)

34L25 Scattering theory

Abstract: A modification of the Kronig--Penney model consisting of
equidistantly spaced $\,\delta'$--interactions is considered. We
prove that absolutely continuous spectrum of such a system disappears
under influence of an external electric field. The result extends to
periodic lattices of non--identical $\delta'$ interactions and
potentials whose behavior, up to a bounded term, is powerlike with
a power $\,\mu\le 1\,$ on the decreasing side.

Keywords: one dimensional Schroedinger operator, Kronig--Penny model, absolutely continuous spectrum