Indefinite Sturm-Liouville operators $ (\sgn x) (- \frac{d^2}{dx^2} +q(x))$ with finite-zone potentials

Abstract

The indefinite Sturm-Liouville operator $A = (\sgn x)(-d^2/dx^2+q(x))$ 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 $-d^2/d{x}^2+q(x)$ with a finite-zone potential $q$. Jordan structure of the operator $\Adisc$ is described. We present an example of the operator $A = (\sgn x)(-d^2/dx^2+q(x))$ such that $A$ is nondefinitizable and $A$ is similar to a normal operator.

59 pages, LaTex 2e, Version 3, a mistake in Corollary 5.6 has been corrected, the format of pages has been changed

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