Geodesic
Inverse spectral problem for integrodifferential Sturm--Liouville operators with discontinuity conditions
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 427-458

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We consider the Sturm–Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodifferential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and sufficient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem. 
@article{CMFD_2018_64_3_a0,
     author = {S. A. Buterin},
     title = {Inverse spectral problem for integrodifferential {Sturm--Liouville} operators with discontinuity conditions},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {427--458},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a0/}
}
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S. A. Buterin. Inverse spectral problem for integrodifferential Sturm--Liouville operators with discontinuity conditions. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 427-458. http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a0/