A representation for diffusion equation with discontinuous coefficient (Q2823102)

In this study, the existence of a transformation operator for the initial value problem NEWLINE\[NEWLINE\begin{gathered} -y''+ [2\lambda p(x)+ q(x)]\,y= \lambda^2\rho(x)\,y\quad\text{for }0\leq x\leq\pi,\\ y(0,\lambda)= 1,\quad y'(0,\lambda)= w_\nu\lambda,\end{gathered}NEWLINE\]NEWLINE where \(p\in W^1_2(0,\pi)\), \(q\in L_2(0,\pi)\), \(w_\nu= (-1)^{\nu+1}i\), \(\nu=1,2\), NEWLINE\[NEWLINE\rho(x)= \begin{cases} 1\quad &\text{for }0\leq x\leq a,\\ \alpha^2\quad &\text{for }a<x\leq\pi\end{cases},\quad 0<\alpha<1,NEWLINE\]NEWLINE is investigated. For this aim, the authors first state the problem as an integral equation and then prove the existence of an transformation operator and the uniqueness of the solution.