Geodesic
On the limit behaviour of the spectrum of a model problem for the Orr--Sommerfeld equation with Poiseuille profile
Izvestiya. Mathematics , Tome 66 (2002) no. 4, pp. 829-856

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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. 
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     title = {On the limit behaviour of the spectrum of a model problem for the {Orr--Sommerfeld} equation with {Poiseuille} profile},
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S. N. Tumanov; A. A. Shkalikov. On the limit behaviour of the spectrum of a model problem for the Orr--Sommerfeld equation with Poiseuille profile. Izvestiya. Mathematics , Tome 66 (2002) no. 4, pp. 829-856. http://geodesic.mathdoc.fr/item/IM2_2002_66_4_a6/