B.M. Brown, M. Langer, M. Marletta, C. Tretter and M. Wagenhofer,
Eigenvalue enclosures and exclosures for non-self-adjoint problems
in hydrodynamics,
LMS J. Comput. Math. 13 (2010), 65–81
Abstract:
In this paper we present computer-assisted proofs of a number of results in
theoretical fluid dynamics and in quantum mechanics.
An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures
and exclosures for non-self-adjoint boundary eigenvalue problems,
the eigenvalues of which are highly sensitive to perturbations.
We apply the algorithm to: the Orr–Sommerfeld equation with Poiseuille profile
to prove the existence of an eigenvalue in the classically unstable region for
Reynolds number
R = 5772.221 818;
the Orr–Sommerfeld equation with Couette profile to prove upper bounds for the
imaginary parts of all eigenvalues for fixed R and wave number α;
the problem of natural oscillations of an incompressible inviscid fluid in the
neighbourhood of an elliptical flow to obtain information about the
unstable part of the spectrum off the imaginary axis;
Squire’s problem from hydrodynamics; and resonances of
one-dimensional Schrödinger operators.