##
**The Ekman flow and related problems: spectral theory and numerical analysis.**
*(English)*
Zbl 1058.34025

An eigenvalue problem arising in the study of the stability of the Ekman boundary layer is of the form
\[
M\zeta = \lambda N \zeta, \quad \zeta =(y,z)^T,\quad \lambda \in \mathbb{C},
\]
where \(M,N\) are matrices whose entries are differential expressions of order up to 4 on \([0,\infty),\) and boundary conditions at \(0\) and \(\infty.\) The authors investigate various aspects of the spectral analysis of the operator \(L=M-\lambda N,\) where \(M,N,L\) are defined on subspaces of Sobolev spaces which give precise meaning to the formal boundary conditions. The essential spectrum is defined to be the set \(\sigma_{\text{ess}}(M,N) = \{ \lambda \in \mathbb{C}: M-\lambda N \text{ is not Fredholm}\}\) and in one of the main results of the paper the essential spectrum is located. There follows a detailed analysis of \(L^2\) solutions of related first-order systems of differential equations and this leads to the development of a Titchmarsh-Weyl coefficient \(M(\lambda)\) and an analysis of the convergence of approximations to the spectrum arising from regular problems. The theory is illustrated by numerical results.

Reviewer: W. D. Evans (Cardiff)

### MSC:

34B20 | Weyl theory and its generalizations for ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

34L16 | Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators |

34L05 | General spectral theory of ordinary differential operators |