An inverse problem for differential operator pencils (Q2710740)

The following boundary problem on the half-axis is considered NEWLINE\[NEWLINE y''+(\rho^2+i\rho q_1(x)+\rho_0(x))y=0, \quad x>0,\qquad y'(0)+(i\rho h_1+h_0)y(0)=0, NEWLINE\]NEWLINE where \(\rho\) is the spectral parameter, \(q_j(x)\) are complex-valued functions, \(h_j\) are complex numbers, \(h_1\not=\pm 1\). NEWLINENEWLINENEWLINEThe author establishes (rather complicated) conditions on a function to be the Weyl function to this problem and gives a procedure for recovering the functions \(q_j(x)\) from the given Weyl function. See \textit{M. Jaulent} and \textit{C. Jean} [The inverse \(s\)-wave scattering problem for a class of potentials depending on energy, Commun. Math. Phys. 28, 177-220 (1972)], \textit{M. Jaulent} [Inverse scattering problems in absorbing media, J. Math. Phys. 17, No. 7, 1351-1360 (1976)], \textit{D. H. Sattinger} and \textit{J. Szmigielski} [Differ. Integral Equ. 8, No. 5, 945-959 (1995; Zbl 0831.35152) and Inverse Probl. 12, No. 6, 1003-1025 (1996; Zbl 0863.35074)], and \textit{T. Aktosun, M. Klaus} and \textit{C. van der Mee} [Wave scattering in one dimension with absorption, J. Math. Phys. 39, No. 4, 1957-1992 (1998) and in: Ramm, Alexander G. (ed.), Spectral and scattering theory. Proceedings of the 1st international congress of the International Society for Analysis, Applications and Computing (ISAAC), University of Delaware, Newark, DE, USA, June 3-7, 1997. New York, NY: Plenum Press, 1-18 (1998; Zbl 0903.34073)] for the complete collection.