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Inverse problems for differential operators with singular boundary conditions. (English) Zbl 1090.34007

The authors study inverse spectral problems for differential operators generated by differential equations of the form

-(p 2 (t)z ' (t)) ' +p 1 (t)z(t)=λp 0 (t)z(t)

considered on a finite interval t(t 0 ,t 1 ) and with the boundary conditions at the end point of the interval. The specific feature of the study is that the coefficients p k (t) are assumed to have singularities at t 0 and t 1 , which requires the boundary conditions to be properly understood. In the paper, the case of the so-called regular singularities is treated, which is characterized by the fact that the Liouville transformation applied to the differential equation leads to the Sturm-Liouville equation on a finite interval with a potential having quadratic singularities at the end points.

The authors study three inverse problems, where the operator is to be recovered by using either the Weyl function or the spectral data or two spectra. The corresponding uniqueness theorems are proven, and relations among the different spectral characteristics are established. A constructive procedure for recovering the operator from the spectral data is given.

MSC:
34A55Inverse problems of ODE
34B24Sturm-Liouville theory
34L40Particular ordinary differential operators
47E05Ordinary differential operators