Green's function and Carleman's formula for transmission problems (Q2091161)

The author deals with the problem \begin{gather*} -p(x)u''+q(x)u=\lambda u(x),\quad x\in[-1,0)\cup(0,1],\\ (\ln y)'(-1)=a_1, \quad (\ln y)'(1)=a_2,\\ y(0^+)-y(0^-)=b_1u(0^-),\quad y'(0^+)-y'(0^-)=b_2u'(0^-), \end{gather*} where \[ p(x)=\begin{cases} p_1>0 &\text{for } x\in[-1,0) \\ p_2>0 &\text{for } x\in(0,1], \end{cases} \] \(q\) is real-valued continuous function on \([-1,0)\) and \((0,1]\), \(a_1,a_2,b_1,b_2\) are real constants and \(\lambda\) is a spectral parameter so that \(a_1\neq 0\), \(a_2\neq 0\), \(b_1>-1\), \(b_2>-1\). In particular, the author constructs the Green's function of the problem and introduces a series expansion of eigenfunctions, Parseval's equality and Carleman's formula.