On the limit behaviour of the spectrum of a model problem for the Orr–Sommerfeld equation with Poiseuille profile

This paper deals with a problem on the limiting behaviour of the spectra of the operators $L(\varepsilon)=i\varepsilon y^{\prime\prime}+x^2y$ with Dirichlet boundary conditions on a finite interval as the positive parameter $\varepsilon$ tends to zero. It is proved that the spectrum is concentrated along three curves in the complex plane. These curves connect a knot-point $\lambda_0$, which lies in the numerical range of the operator, with the points 0, 1 and $-i\infty$. We find uniform (with respect to $\varepsilon$) quasiclassical formulae for the distribution of the eigenvalues along these curves.