Functional Model of a Class of Non-selfadjoint Extensions of Symmetric Operators

Wave operators, similarity and dynamics for a class of Schrödinger operators with generic non-mixed interface conditions in 1D

“…It has been shown, indeed, that a key point in the development of the scattering theory for the possibly non-selfadjoint pair {H 0 , H 1 } is the existence of the strong limit on the real axis of the characteristic functions associated with H i=0,1 (e.g. in [29] and [30]). In particular, the Theorem 4.1 in [29] makes use of this assumption to study the existence of the related wave operators.…”

Functional Model of a Closed Non-Selfadjoint Operator

“…It is convenient to use representation for A in the form (2.3) for z ∈ C − and in the form (2.4) for z ∈ C + . The interested reader is referred to [44] Theorem 2.2, where analogous calculations were carried out for a special case of operator A.…”

“…It was already successfully applied (without a proof) to the study of nondissipative operators from a fairly wide class in [42,43], where the notion of local absolutely continuous and singular subspaces was examined and utilized for the subsequent study of scattering theory for a pair of nonselfadjoint operators. Owing to the generic form of operator under consideration, results obtained in the current paper cover both cases of the model for nonselfadjoint additive perturbations [28] and for extensions of symmetric operators [44]. As a direct consequence, considerations below can be used for study of selfadjoint operators subject to a nonselfadjoint additive perturbations in combination with nonselfadjoint boundary conditions.…”

“…From the applications perspective, the generic answer given by [25] is not quite satisfactory. In particular, it does not offer much help for the model representation in situations like nonselfadjoint extensions of concrete symmetric operators [44], or in similar cases where the operator is not easily representable as a sum of its real and imaginary parts as required by (1.1). Slightly more general settings, such as operators of typical boundary value problems recently studied on the abstract level in [45], are not covered by arguments given in [28,44], and the results of [25] are not easily translatable into the problem's terms either.…”

“…However, efforts were made in order to clarify the argumentation and to provide the reader with references to the literature where analogous calculations can be found. Two main resources in this regard are the original work [28] and the recent paper [44] where all relevant computations are supplied in full length. For properties of Potapov-Ginzburg transform and the concise account of results from [5] we refer the reader to the book [6].…”