The boundary value problem \(L=L(Q(x),\rho,\beta)\) of the following form is considered: \[ -y''+Q(x)y=\lambda\rho(x) y, \quad 0<x<\pi, \tag{1} \]
\[ y'(0)=y'(\pi)=0, \tag{2} \]
\[ y\Big(\frac\pi2+0\Big)=\beta y\Big(\frac\pi2-0\Big), \quad y'\Big(\frac\pi2+0\Big)=\beta^{-1} y'\Big(\frac\pi2-0\Big), \tag{3} \] where \(Q\) is a real-valued function, \(Q\in L_2(0,\pi);\) \[ \rho(x)=\begin{cases} 1, & x<\frac\pi2,\\ \alpha^2, & x>\frac\pi2, \end{cases} \quad 0<\alpha<1, \] and \(\beta>0.\)
The uniqueness of a solution of the following half-inverse spectral problem is studied: given the spectrum of \(L,\) find \(L\) under the assumption that \(Q(x)\) is known on \((0,\pi/2)\). The corresponding uniqueness result was obtained earlier in [C.-T. Shieh and V. A. Yurko, J. Math. Anal. Appl. 347, No. 1, 266–272 (2008; Zbl 1209.34014)].