Indefinite Sturm-Liouville operators (sgn x)(-d2/dx2+q(x)) with finite-zone potentials

Abstract

The indefinite Sturm-Liouville operator A = (sgn x)(-d 2 /dx 2 + q) is studied. It is proved that similarity of A to a selfadjoint operator is equivalent to integral estimates of Cauchy type integrals. Some simple sufficient and necessary conditions for the similarity to a selfadjoint operator in terms of Weyl functions are given. For operators with a finite-zone potential q , the components A ess and A disc of A corresponding to the essential and the discrete spectrums, respectively, are investigated. The main result of the paper is a criterion of similarity of the operator A (resp. A ess ) with a finite-zone potential q to a normal (resp. selfadjoint) operator. It is given in terms of the Weyl functions corresponding to the Sturm-Liouville operator -d 2 /dx 2 + q. Jordan structure of the operator A disc is described. An example of a non-definitizable operator A that is similar to a normal operator is presented too.