Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions
Copyright policy )Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions
In this paper, we for the first time get constructive solution for the inverse Sturm-Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction.
- Samara State Aerospace University Russian Federation
- Saratov State University Russian Federation
- Peoples' Friendship University of Russia Russian Federation
polynomials in the boundary conditions, Inverse problems involving ordinary differential equations, Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.), uniqueness theorem, Mathematics - Spectral Theory, Sturm-Liouville theory, Sturm-Liouville operator, constructive algorithm, FOS: Mathematics, inverse spectral problems, Boundary eigenvalue problems for ordinary differential equations, Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter, Spectral Theory (math.SP), singular potential
polynomials in the boundary conditions, Inverse problems involving ordinary differential equations, Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.), uniqueness theorem, Mathematics - Spectral Theory, Sturm-Liouville theory, Sturm-Liouville operator, constructive algorithm, FOS: Mathematics, inverse spectral problems, Boundary eigenvalue problems for ordinary differential equations, Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter, Spectral Theory (math.SP), singular potential
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