The Operator (sgn x) d 2 dx 2 is Similar to a Selfadjoint Operator in L 2 ( R )

Abstract

Krein space operator-theoretic methods are used to prove that the operator (sgn x) d is similar to a selfadjoint operator in the Hilbert space L2 (R) . Let L be a symmetric ordinary differential expression. Spectral properties of the operators associated with the weighted eigenvalue problem Lu = )wu have been studied extensively. When w is positive, this problem leads to a selfadjoint problem in the Hilbert space L2(w). In recent years there has been considerable interest in the case when w changes sign; for a survey see [5] and also [2]. In this case the problem may have nonreal and nonsemisimple spectrum. Since the problem is symmetric with respect to an indefinite scalar product, it is natural to consider the problem in the associated Krein space. The corresponding operator can be studied using the spectral theory of definitizable selfadjoint operators in Krein spaces. For definitions and basic results of this theory see [4]. Let (X, [*I*]) be a Krein space, A a definitizable operator in X, and E the spectral function of A. Of particular interest are the so-called critical points of A where its spectral properties are different from the spectral properties of a selfadjoint operator in Hilbert space. Definitizable operators may have at most finitely many critical points. Significantly different behavior of the spectral function occurs at singular critical points in any neighborhood of which the spectral function is unbounded. The critical points which are not singular are called regular. The simplest class of definitizable operators are positive operators with nonempty resolvent set: an operator A is positive if [Ax, x] > 0, x E -'(A). The spectrum of a positive operator is real; only 0 may be a nonsemisimple eigenvalue; only 0 and x may be critical points. Moreover, if 0 and x are not singular critical points and if 0 is not an eigenvalue, then the operator A is a selfadjoint operator in the Hilbert space (X, [(E(R+) E(R_)) Ii); see [4, Theorem 5.7]. Received by the editors June 15, 1993 and, in revised form, July 1, 1993. 1991 Mathematics Subject Classification. Primary 47B50, 47E05; Secondary 47B25, 34L05. The second author was supported in part by Fond za znanstveni rad Hrvatske. This research was done while the second author was visiting the Department of Mathematics, Western Washington University. Q 1995 American Mathematical Society 0002-9939/95 $1.00 + S.25 per page