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Inverse problem having special singularity type from two spectra. (English) Zbl 1300.34039

Summary: It is well-known that the two spectra \(\{\lambda_n\}\) and \(\{\mu_n\}\) uniquely determine the potential function \(q(x)\) in a Sturm-Liouville equation defined on the unit interval and having the special singularity \(q(x)={\delta\over x^p}+ q_0(x)\) (where \(\delta\) is an constant, \(1<p<2\)) at the point zero.
In this work, we give the solution of the inverse problem on two partially non-coinciding spectra for the Sturm-Liouville equation with to peculiarity at zero. In particular, we obtain Hochstadt’s theorem concerning the structure of the difference \(q(x)-\widetilde g(x)\).

MSC:

34A55 Inverse problems involving ordinary differential equations
34B24 Sturm-Liouville theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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