Perturbation of Eigenfunction Expansions

Abstract

The present note is concerned with an operator-theoretical approach to the construction of eigenfunction expansions by the perturbation method and its applicatioln to the study of n-dimensional Schrddinger operators.1 Our result on Schrbdinger operators generalizes a result of Ikebe2 for n = 3 to an arbitrary n. The abstract part of the results will be stated for a simplified situation, which, however, would cover various problems of practical interest. A full account with a complete proof will be published later.1. Let 15 be a separable Hilbert space with the inner product denoted by (u,v).We denote by B the set of all bounded linear operators from ID into I, and by C., c B the set of all completely continuous operators.Let Ho be a self-adjoint operator in &, A lie in B with the range being dense in &, and B be a densely defined linear operator in A. The following conditions will be assumed throughout §1-3:exist for any real X and Q(z) is continuous in 40 either the upper or the lower half-plane, including the corresponding bank of the real axis.Under assumptions (I) and (II) it can be shown that H1 = Ho + AB is a selfadjoint operator with domain equal to TD(Ho). For the pair (HoH1), the following theorem was proved in a previous work of the writer.4 THEOREM 1. Under the above situation we have: (a) Ho is absolutely continuous; (b) the set es of all real X for which at least one of I + Q(X i iO) is not invertible is a closed set with the Lebesgue measure 0; (c) the singular spectrum of H1 is contained in e,; and (d) the absolutely continuous (abbr. a.c.) part H1,,ac of H1 is unitarily equivalent to Ho.It follows from assumption (III) that M0(X) = s-lim (2iri)-'To(, + iE) . 0(1) e 1 0 exists for a.e. X (s-lim means the strong limit of operators). If M0(X) exists for every X in a closed interval a, and if Mo(X)'1' is Holder continuous in l with H6lder exponent greater than 1/2, then it can be shown (cf. ref. 4, §7) that e fnl is a finite set, each point being an eigenvalue of H1 with finite multiplicity. 2. Let 9? be the range of A which is assumed to be dense in ,A. Let [A ] denote the restriction of A to the orthogonal complement 9)1(A) of the null space of A and introduce an inner product (u,v) into T by (u,v) = ( [A ]-1u, [A ]-1v). This inner product makes T a Hilbert space whose topology is stronger than the one induced by the norm of SA. Thus, we have canonically T c S c 9?', 9' being the dual space of 9?. We use this triplet of spaces to formulate the generalized eigenfunction expansion along the lines described by Gelfand, Silov, and others.5 Here and sub-1213

Wave and Scattering Operators for Second-Order Elliptic Operators in $R^3$

“…Note that in the present case use has to be made of (5. 8) It is known (see Ikebe [1] and Kuroda [4]) that if q(x) behaves asymptotically like x ~2~s, and if L = ( -A-f #)o, then W±(L, L 0 ) exist and are complete, though the assumption on q(x) is not precisely stated (see (C. VI) below). Note that the rate of decrease 0(\x\ ~2~s) for q(x) is in a sense milder than the requirement that q^L ly because the latter claims that q(x) vanish like \x\~z~* at oo if it decreases to some negative power of x\.…”

“…Recently, Kuroda and Thoe's result has been extended to the Schrodinger operator with external magnetic field by Ushijima [6]. (In [4], [5] and [6] are considered the Schrodinger operators in n dimensions, while we have restricted ourselves in this paper to the case w = 3.) Our result, however, differs from [1], [4], [5] and [6] in that L -L 0 , the perturbation, is a second-order differential operator, whereas it is at most a first-order one in the latter works.…”