Geodesic
Non-selfadjoint singular perturbations and spectral properties of the~Orr--Sommerfeld boundary-value problem
Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 137-156

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A new approach to the analysis of the asymptotic behaviour (and in particular, of the degree of non-orthogonality) of the eigenfunctions and associated functions of non-selfadjoint singularly perturbed operators and boundary-value problems is suggested; the main attention is paid to the case when the spectrum fails to be lower semicontinuous under singular perturbations. As a model case of the transition from a discrete to a continuous spectrum a Sturm–Liouville problem with a small parameter multiplying the second derivative is considered. Spectrum localization is studied and the growth of the degree of non-orthogonality of the system of eigenfunctions and associated functions of the Orr–Sommerfeld problem as the viscosity vanishes is established. 
@article{SM_1997_188_1_a6,
     author = {S. A. Stepin},
     title = {Non-selfadjoint singular perturbations and spectral properties of {the~Orr--Sommerfeld} boundary-value problem},
     journal = {Sbornik. Mathematics},
     pages = {137--156},
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     volume = {188},
     number = {1},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_1_a6/}
}
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S. A. Stepin. Non-selfadjoint singular perturbations and spectral properties of the~Orr--Sommerfeld boundary-value problem. Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 137-156. http://geodesic.mathdoc.fr/item/SM_1997_188_1_a6/