One-dimensional Schrödinger operators withδ′-interactions on Cantor-type sets

Abstract

Abstract. We introduce a novel approach for defining a δ ′ -interaction on a subset of the real line of Lebesgue measure zero which is based on SturmLiouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ ′ -interactions concentrated on sets of complicated structures.

Schrödinger evolution of superoscillations with $$\delta $$- and $$\delta '$$-potentials

“…where we used the coefficients in the power series representation (17) of Λ. In a similar way as in (27) and (29) for the free part, we can use the binomic formula to further expand this series into the form (9). Hence we get an operator U δ,+ (t, x) :…”

“…A detailed discussion of Schrödinger operators with δ and δ ′ -point interactions can be found in the standard monograph [13]. For further reading on Schrödinger operators with singular potentials we refer the reader to, e.g., [16,17,24,25,26,29,30,31,32,33,38,40,41,42,43,44] and the references therein.…”

On continuous dependence of solutions of dynamic equations

“…It can be also useful in the case when the construction of T is based on the Kuratowski limit of sets (like the Cantor set K -cf. [23,33] or [13]) -see Section 4. Note that some estimation related to this result can be also found in [1,Appendix].…”

“…For an initial value problem on a Cantor set there is no ''approximated'' difference equations, but the solution can be obtained as a limit of solutions on a sequence of time scales convergent to the Cantor set. Recall that the problems on Cantor sets are not only investigated as an example of dynamic problems -see the recent work [13], for instance. Let us also recall that in [31] many linear equations on discrete, but not necessarily compact time scales, are investigated.…”

δ′-interaction as a limit of a thin Neumann waveguide with transversal window

“…The first rigorous treatment of δ -interactions was made in [17]. Further investigations were carried out in [3,4,8,9,13,14,18,19,25,26,30,31,37] (see also [6] for the multidimensional version of δ -interactions).…”