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 Mat. Sb. (N.S.), 1970, Volume 81(123), Number 4, Pages 525–551 (Mi msb3384)

Stability of the problem of recovering the Sturm–Liouville operator from the spectral function

V. A. Marchenko, K. V. Maslov

Abstract: We consider a differential operator $\mathscr L=(h,q(x))$ generated by a Sturm-Liouville operation $l[y]=-y"+q(x)y$ on the linear manifold of finite twice-differentiable functions $y(x)$ satisfying the boundary condition $y'(0)-hy(0)=0$. Let $\rho(\mu)$ be the spectral function of this operator. From $\rho(\mu)$, as is well known, we can recover the operator $\mathscr L$, i.e. the number $h$ and the function $q(x)$. Let $V_\alpha^A$ be the set of operators $\mathscr L$ for which
$$|h|\leqslant A,\qquad\int_0^x|q(t)| dt\leqslant\alpha(x)\quad(x<0<\infty).$$

We now investigate how much information about the operator $\mathscr L\in V_\alpha^A$ can be obtained if its spectral function $\rho(\mu)$ is known only for values of $\mu$ on a finite interval.
In the present article we obtain estimates for the difference in the potentials $q_1(x)-q_2(x)$, in the boundary parameters $h_1-h_2$ and in the solutions of the corresponding differential equations under the condition that the spectral functions of the two operators in $V_\alpha^A$ coincide on a finite interval.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1970, 10:4, 475–502

Bibliographic databases:

UDC: 517.43
MSC: 34B24, 47E05, 34L05, 47G20, 45J05, 34Dxx

Citation: V. A. Marchenko, K. V. Maslov, “Stability of the problem of recovering the Sturm–Liouville operator from the spectral function”, Mat. Sb. (N.S.), 81(123):4 (1970), 525–551; Math. USSR-Sb., 10:4 (1970), 475–502

Citation in format AMSBIB
\Bibitem{MarMas70} \by V.~A.~Marchenko, K.~V.~Maslov \paper Stability of the problem of recovering the Sturm--Liouville operator from the spectral function \jour Mat. Sb. (N.S.) \yr 1970 \vol 81(123) \issue 4 \pages 525--551 \mathnet{http://mi.mathnet.ru/msb3384} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=264154} \zmath{https://zbmath.org/?q=an:0195.43301|0216.17102} \transl \jour Math. USSR-Sb. \yr 1970 \vol 10 \issue 4 \pages 475--502 \crossref{https://doi.org/10.1070/SM1970v010n04ABEH002160} 

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This publication is cited in the following articles:
1. A. M. Savchuk, A. A. Shkalikov, “Inverse Problems for Sturm–Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability”, Funct. Anal. Appl., 44:4 (2010), 270–285
2. Emrah Yilmaz, Sertac Goktas, Hikmet Koyunbakan, “On the Lipschitz stability of inverse nodal problem for p-Laplacian Schrödinger equation with energy dependent potential”, Bound Value Probl, 2015:1 (2015)
3. Manafov Manaf D. Z. H. Kablan A., “Inverse Spectral and Inverse Nodal Problems For Energy-Dependent Sturm-Liouvillee Quations With Delta-Interaction”, Electron. J. Differ. Equ., 2015, 26
4. A. M. Savchuk, “Reconstruction of the Potential of the Sturm–Liouville Operator from a Finite Set of Eigenvalues and Normalizing Constants”, Math. Notes, 99:5 (2016), 715–728
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