Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

Abstract

For a real-valued function V from the Faddeev-Marchenko class, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the formUnder certain conditions the functions ε −2 V (x/ε) converge in the sense of distributions as ε → 0 to δ ′ (x), and then the limit S 0 of S ε might be considered as a "physically motivated" interpretation of the one-dimensional Schrödinger operator with potential δ ′ .

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

“…The particular cases B = aδ(x) and B = bδ ′ (x), a, b ∈ R, are then studied in detail. We show that they yield families of Schrödinger operators that coincide (exactly in the first case, and to a large extend in the second case) with the families of norm resolvent limit operators that were obtained in [29,30].…”

“…As we have already mentioned in the introduction, one possible interpretation of H = H 0 + B is that it stands for the norm resolvent limit of a sequence of operators H n = H 0 + B n , where B n = B n · and B n is a sequence of regular potentials such that B n −→ B in D ′ . The case B = aδ(x)+bδ ′ (x), with a, b ∈ R, has been extensively studied in the literature (see [30,49] and the references therein). It turns out that for B n −→ B = aδ(x) in D ′ , the norm resolvent limit of H n is (for a large class of regular potentials B n ) (6.1)…”

“…The books [2,4,37] and the papers [6,7,11,26,30,39,42,44,46,48] provide a detailed discussion and an extensive list of references on this subject. In the present work we will address the problem, that emerges naturally in this context, of constructing a boundary potential formulation of the operators L that is entirely defined in terms of standard Schwartz distributions.…”

“…in the norm resolvent sense) of sequences of operators of the form (1.2) H n = H 0 + B n , B n = B n · where B n belongs to some space of regular functions and satisfies B n −→ B in D ′ . Sequences of this kind have been used in many applications, (see, for instance, [30] and the references therein), and their convergence properties were studied for particular cases [1,15,16,18,30,44,47,48]. Unfortunately, the relation between the convergence of B n in D ′ and the convergence of the associated operators H n in the norm resolvent is not straightforward [12,28,29,48,49].…”

“…The case B = δ ′ was studied in detail. It was found that the formal operator H 0 + δ ′ stands (in the sense above) for, at least, a one-parameter family of distinct s.a. Schrödinger operators with a point interaction [30,44,49]. An interesting problem is then whether the operators H 0 + B admit an intrinsic formulation in terms of distributions and also, whether such formulation coincides with the norm resolvent limit of suitable sequences of Schrödinger operators with regular potentials.…”