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Indefinite Sturm–Liouville operators $$(\operatorname{sgn} x)(-\frac{d^2}{dx^2}+q(x))$$ with finite-zone potentials. (English) Zbl 1146.47032
The paper studies the indefinite Sturm-Liouville operator, $$A= (\operatorname {sgn} x)(-d^2/dx^2+q)$$. It is proven that similarity of $$A$$ to a selfadjoint operator is equivalent to integral estimates of Cauchy type integrals. Some simple sufficient and necessary conditions are given for the similarity to a selfadjoint operator in terms of Weyl functions. For operators with a finite-zone potential $$q$$, the components $$A_{\text{ess}}$$ and $$A_{\text{disc}}$$ of $$A$$ corresponding to the essential and the discrete spectra, respectively, are investigated. The main result of the paper is a criterion for the similarity of the operator $$A$$ (resp., $$A_{\text{ess}}$$) 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, $$-d^2/dx^2+q$$. The Jordan structure of the operator $$A_{\text{disc}}$$ is described. An example of a non-definitizable operator $$A$$ that is similar to a normal operator is also presented.

##### MSC:
 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX) 34B24 Sturm-Liouville theory 34B09 Boundary eigenvalue problems for ordinary differential equations 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 47B50 Linear operators on spaces with an indefinite metric
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