Open AccessInverse Spectral Problem for Integrodifferential Sturm-Liouville Operators with Discontinuity Conditions
Author(s) -
Sergey Buterin,
Sergey Buterin
Publication year - 2018
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2018-64-3-427-458
Subject(s) - mathematics , sturm–liouville theory , uniqueness , discontinuity (linguistics) , operator (biology) , mathematical analysis , boundary value problem , inverse , inverse problem , singularity , convolution (computer science) , spectrum (functional analysis) , interval (graph theory) , biochemistry , chemistry , physics , geometry , repressor , quantum mechanics , combinatorics , machine learning , artificial neural network , transcription factor , computer science , gene
We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a nite 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 integrodierential 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 sucient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.