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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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