Journal of Mathematical Physics, Analysis, Geometry
2021, vol. 17, No 1, pp. 95-115   https://doi.org/10.15407/mag17.01.095     ( to contents , go back )
https://doi.org/10.15407/mag17.01.095

On a Spectral Inverse Problem in Perturbation Theory

V.A. Marchenko

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: marchenko@ilt.kharkov.ua

A.V. Marchenko

Moody's Analytics, 5001 Yonge Street, Siute 1300, Box 172, Toronto, Ontario, M2N 6P6, Canada
E-mail: avmarch@mail.com

V.A. Zolotarev

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: zolotarev@ilt.kharkov.ua

Received February 20, 2021.

Abstract

We consider an inverse spectral problem for Sturm–Liouville operators $% \widehat{H}_{V}$ defined on the interval $\left[ a,b\right] $ by a certain potential $V \in L^{2}[a,b]$ and mixed separated boundary conditions. We show that if the $L^{1}$-norm of $V$ is small enough, then there exists $V_{\text{app}}$ such that $\|V-V_{\text{app}}\|_{L^{2}}=O(\|V\|^2_{L^{1}})$ and we indicate an algorithm to find $V_{\text{app}}$. The algorithm determines the Fourier coefficients of $V_{\text{app}}$ with respect to eigenfunctions $\{\psi _{k,0}\}_{k=1}^{\infty }$ of the unperturbed operator $\hat{H}_{0}$ via eigenvalues $\{\lambda _{k,V}\}_{k=1}^{\infty }$ of the "perturbed" operator $\hat{H}_{V}$, the values of its eigenfunctions $\{\psi _{k,V}\}_{k=1}^{\infty }$ at the endpoints of $[a,b]$, and values of $\{\psi _{k,V}\}_{k=1}^{\infty }$ and their derivatives at the middle of $[a,b]$.

Mathematics Subject Classification 2010: 34A55, 34B24, 47A55
Key words: spectral theory, potential, inverse problem, perturbation theory

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