A model of transition from discrete spectrum to continuous one in the singular perturbation theory

The spectral problem
\begin{gather*} i\varepsilon y''(x)+(x-\lambda)y(x)=0, \\ y(-1)=y(1)=0 \end{gather*}
is considered where $\lambda$ is a spectral parameter and $\varepsilon>0$ is a small parameter. Spectrum localization, behavior of eigenfunctions and Green function of this problem are studied by analytical means.