On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

A model spectral problem of the form $-i\varepsilon y''+xy=\lambda y$ on the finite interval $[-1,1]$ with the Dirichlet boundary conditions is considered. Here $\lambda$ is the spectral parameter and $\varepsilon$ is positive. The behavior of the spectrum of this problem as $\varepsilon\to 0$ is completely investigated. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.