A uniqueness theorem for Sturm-Liouville equations with a spectral parameter rationally contained in the boundary condition (Q446195)

The authors consider the boundary value problem NEWLINE\[NEWLINE -y''+q(x)y=\lambda y, \quad 0<x<\pi, \tag{1} NEWLINE\]NEWLINE NEWLINE\[NEWLINE y'(0)-hy(0)=0, \tag{2} NEWLINE\]NEWLINE NEWLINE\[NEWLINE \theta_2(\lambda)y'(\pi)-\theta_1(\lambda)y(\pi)=0, \tag{3} NEWLINE\]NEWLINE where \(q(x)\) is a real-valued function, \(q(x)\in L_2(0,\pi),\) \(h\in{\mathbb R},\) \(\theta_1(\lambda)\) and \(\theta_2(\lambda)\) are mutually-prime polynomials with real coefficients. They study an inverse problem: given the spectral characteristics (the spectrum of the boundary value problem (1)--(3) along with the coefficients of the principal parts of its Weyl function), find \(q(x)\), \(h\) and \(\theta_1(\lambda)/\theta_2(\lambda)\). The uniqueness theorem of this inverse problem is proved.NEWLINENEWLINEThis uniqueness theorem together with a reconstructive procedure were obtained earlier for a more general case (both the boundary conditions (2), (3) are polynomially dependent on the spectral parameter, and all coefficients of (1)--(3) including \(q\) are complex-valued) by \textit{G. Freiling} and \textit{V. A. Yurko} [Inverse Probl. 26, No. 5, Article ID 055003 (2010; Zbl 1207.47039)].