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Stability indices for constrained self-adjoint operators

Authors: Todd Kapitula and Keith Promislow
Journal: Proc. Amer. Math. Soc. 140 (2012), 865-880
MSC (2010): Primary 35P05, 47A53, 47A75
Published electronically: July 5, 2011
MathSciNet review: 2869071
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Abstract: A wide class of problems in the study of the spectral and orbital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called ``energy spectrum'', that is, the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert space. We present a substantially simplified proof of the negative eigenvalue count for such constrained, self-adjoint operators, and extend the result to include an analysis of the location of the point spectra of the constrained operator relative to that of the unconstrained operator. The results are used to provide a new proof of the Jones-Grillakis instability index for generalized eigenvalue problems of the form $ (\mathcal{R}-z\mathcal{S})u=0$ via a careful analysis of the associated Krein matrix.

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Additional Information

Todd Kapitula
Affiliation: Department of Mathematics and Statistics, Calvin College, Grand Rapids, Michigan 49546

Keith Promislow
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Received by editor(s): September 21, 2010
Received by editor(s) in revised form: October 25, 2010, and December 10, 2010
Published electronically: July 5, 2011
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.