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On the inverse Sturm-Liouville problem. (English) Zbl 0568.65056
The problem of the determination of a function q(x) and numbers h,H\(\in R\) such that the Sturm-Liouville problem \((1)\quad y''+(\lambda - q(x))y=0,\) \(0<x<\pi\), \((2)\quad y'(0)-hy(0)=0,\) \(y'(\pi)+Hy(\pi)=0\) has given spectral characteristics \(\{\lambda_ n,\rho_ n\}^{\infty}_{n=0}\) is considered \((\lambda_ n\) is the n-th eigenvalue and \(\rho_ n=\int^{\pi}_{0}(\phi (x,\lambda_ n))^ 2dx\) where \(\phi (x,\lambda_ n)\) is the eigenfunction satisfying the initial condition \(u(0)=1\), \(u'(0)=h)\). A new proof on the theorem of I. M. Gel’fand and B. M. Levitan [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 309-360 (1951; Zbl 0044.093)] of the solvability of the above inverse problem is presented. Moreover, it is proved that this inverse problem is well-posed, i.e. if two different spectral characteristics of two problems (1) with the same boundary conditions (2) are close in a certain sense, then the difference between the coefficients of equations is small.
Reviewer: T.Reginska

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
34L99 Ordinary differential operators