Equipped absolutely continuous subspaces and stationary construction of the wave operators in the non-self-adjoint scattering theory

Abstract

This paper is devoted to the problem of correct definition of local wave operators (WO) within the context of non-self-adjoint scattering theory for a pair of spaces. The approach proposed in this paper allows us to reduce the problem of investigation of WO for a pair of non-self-adjoint operators to an equiwlent problem of "self-adjoint" theory.From the point of view of spectral operator theory, the mathematical scattering theory in a loose sense is an instrtunent of investigation of the structure of the absolutely continuous subspace of a self-adjoint operator. The abundance of results in the scattering theory (see, for example, [21]) has stimulated interest in various generalizations of the latter. The explicit formulation of propositions of the non-self-adjoint scattering theory inevitably leads to two different (however, related) problems. The first one is to describe some class of operators for which it is possible in a sense to introduce the principal object of the scattering theory -the absolutely continuous subspace. The second one is to define WO for a pair of operators of the chosen class coordinated with the properties of this subspace, and also to find some sufficient conditions for their existence. The present paper continues the investigations of a number of papers [6-8, 11, 13-15, 23-27] that are devoted to the solution of the stated problems in the case of operators in a sense "close" to self-adjoint ones. The notion of "closeness" accepted by us is somewhat wider than in the aforesaid papers: the class under investigation here is distinguished by the assumption that almost everywhere on the real axis there exist boundary values of the so-called characteristic function of the non-self-adjoint operator under investigation. Definitions of absolutely continuous and singular subspaces of such operators are introduced according to the results of [13,14,11]. Some additional properties of these subspaces were investigated in [22]. In Sec. 1 we briefly present some information from [13,14,11,22] that we need-in the sequel. Moreover, the same section contains an interpretation of the absolutely continuous subspace of a non-self-adjoint operator (probably going back to the papers [12,10,19,20]), as an equipped (Hilbert) space. This allows us to give in Sec. 2 a simple definition of WO for a pair of operators from the abovementioned class. This definition is of the same form as the definition of WO given in [4] while developing an abstract stationary approach to the self-adjoint scattering theory. Using the techniques of equipped spaces, we show that the introduced WO coincide up to inessential stipulations with the WO constructed for a certain pair of self-adjoint operators uniquely determined by the initial ones. For this reason, we can directly use the results of [4] when formulating a number of theorems. As in [4], the main assumptions providing the existence of WO are reduced to the requirement of the existence of boundary values (in a suitable sense) for the "bordered" resolvents almost everywhere o...

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

“…As a consequence of Lemma 2.2, we have: 28) and the Picard series uniformly converges to 29) and, taking into account the definition:…”

“…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.…”

“…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.…”

Absolutely continuous and singular subspaces of a non-self-adjoint operator

“…The theorem proved above is crucial to establish stationary representations for wave operators in the non-self-adjoint scattering theory. Unfortunately, by the author's fault, there is a regrettable inaccuracy in the statement of a theorem in [21] Here we consider briefly the problem of definition and localization of the absolutely continuous spectrum of the operator L.…”

Functional Model of a Closed Non-Selfadjoint Operator

“…The paper offers such a model, free of limitations described above. 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].…”