Singular continuous measures in scattering theory

Abstract

Examples are presented of potentials V for which --j + V(r) inZ?(0, oo) has singular continuous spectrum, and the physical interpretation is discussed.

Band-Gap Structure and Singular Character of Bounded One-Dimensional Multibarrier Potentials

“…That is, the incoming wave function remains the same (preserved) after passing the dense array of potential barriers [4,5]. Thus, the three properties which are characterized in the literature [8,9,10,11] as signs of singular systems are all found also in the bounded multibarrier potential discussed here. These properties are : (1) nonvanishing gaps, (2) the ability to pass all the barriers of the system and (3) zero range potential for each barrier which arises here from the fact that the bounded spatial length contains very large number of barriers.…”

“…That is, the gaps in their energy spectrum, which do not depend on any such undefined L and c, can not be preserved, as in the bounded potential, merely by taking some limit or other values of these L and c. Thus, their spectrum is, as remarked, generally absolutely continuous [13]. There are, of course, the exceptional infinite periodic systems [9,10,11] mentioned in Section 5 which are characterized by nonvanishing gaps but for other reasons not related at all to any L and c. The bounded multibarrier potential discussed here allows one not only to define its total length L and the related ratio of its total interval to total width but also to express the location of each barrier [4,5,6] and consequently the allowed energies in terms of L and c. Thus, there may be found, as actually shown in Sections 3-4, specific values of L and c for which the gaps in the energy spectrum do not disappear even for large values of E or (and) large κ. Moreover, for large (and intermediate) L and small c the energy E, as function of κ, of the bounded multibarrier potential have been shown to have the form of quasi-periodic half squares which are jumpy and discontinuous at their vertical sides (see Figures 1-2 and 4-5).…”

“…(23) and also Figure 7 from which we realize that there exist quasi-periodic gaps for small values of L). Now, it is accepted in the literature (see, for example, [9,10]) that if the energy spectrum of any system have gaps which do not diminish for either large values of the energy E or large κ then the relevant system is considered singular. We have shown in Sections 3-4 that not only the energy spectrum of the bounded multibarrier potential have gaps which remain constant along the κ axis but that even for the parts of the epectrum which have bands there are points at which the energy, as function of κ, is discontinuous.…”

“…That is, as written in [9] "the particle will be transmitted at least once through each barrier and then it will return again to the initial point from which it has started". Thus, for these infinite systems the singular continuity is characterized by complete reflection that follows a former stage during which all the barriers are transmitted.…”

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

“…As noted in the introduction, all the examples we present in this paper are obtained by applying the (suitably modified) methods of the diagonal sparse case to the off diagonal case and using theorem 2.4. In particular, theorem 2.2 follows from theorem 2.4 by the methods in [7].…”