Geodesic
Stability of the problem of recovering the Sturm–Liouville operator from the spectral function
Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 475-502 
V. A. Marchenko; K. V. Maslov. Stability of the problem of recovering the Sturm–Liouville operator from the spectral function. Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 475-502. http://geodesic.mathdoc.fr/item/SM_1970_10_4_a1/
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     author = {V. A. Marchenko and K. V. Maslov},
     title = {Stability of the problem of recovering the {Sturm{\textendash}Liouville} operator from the spectral function},
     journal = {Sbornik. Mathematics},
     pages = {475--502},
     year = {1970},
     volume = {10},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_4_a1/}
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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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[6] V. A. Marchenko, “Teoremy tauberova tipa v spektralnom analize differentsialnykh operatorov”, Izv. AN SSSR, seriya matem., 19 (1955), 381–422 | Zbl

[7] V. A. Marchenko, “Razlozhenie po sobstvennym funktsiyam nesamosopryazhennykh singulyarnykh differentsialnykh operatorov vtorogo poryadka”, Matem. sb., 52(94) (1960), 739–788