Sturm - Liouville operators ( sgn x ) ( − d 2 dx 2 + q ( x ) ) with finite - zone potentials

The indefinite Sturm-Liouville operator A = (sgnx)(−d2/dx2 + q(x)) is studied. It is proved that similarity of A to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of Weyl functions are given. For operators with a finite-zone potential, the components Aess and Adisc of A corresponding to essential and discrete spectrums, respectively, are considered. A criterion of similarity of Aess to a selfadjoint operator is given in terms of Weyl functions for the Sturm-Liouville operator −d2/dx2 + q(x) with a finite-zone potential q. Jordan structure of the operator Adisc is described. We present an example of the operator A = (sgn x)(−d2/dx2 + q(x)) such that A is nondefinitizable and A is similar to a normal operator.