A class of inverse problems for a quadratic pencil of Sturm-Liouville operators (Q5930605)

The authors deal with the boundary value problem \(L\): \[ y''+[\lambda^2-2\lambda p(x)-q(x)]y=0, \quad 0<x<\pi, \] \[ by(0)+y'(0)+\omega y(\pi)=0,\quad -\overline\omega y(0)+ay(\pi)+y'(\pi)=0, \] with \(p(x)\in W_2^1[0,\pi], q(x)\in L_2[0,\pi]\), \(a\) and \(b\) are real, and \(\omega\) is a complex number. Properties of the spectrum are studied, and the inverse problem of recovering \(p(x)\), \(q(x)\) and the coefficients of the boundary conditions from the given spectral data is investigated. For this inverse problem a uniqueness theorem is formulated and conditions for the solvability of the inverse problem are obtained.