This article is cited in 18 papers

Brief communications

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

A. V. D'yachenko^a, A. A. Shkalikov<sup>b

a</sup> M. V. Lomonosov Moscow State University
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

Keywords: the Airy function, Couette flow, quasiclassical eigenvalue formulas.

UDC: 517.927+517.928

Received: 29.10.2001

DOI: 10.4213/faa208

 English version: Functional Analysis and Its Applications, 2002, 36:3, 228–232

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