Stability indices for constrained self-adjoint operators

Kapitula, Todd; Promislow, Keith

doi:10.1090/S0002-9939-2011-10943-2

Stability indices for constrained self-adjoint operators
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by Todd Kapitula and Keith Promislow PDF

Proc. Amer. Math. Soc. 140 (2012), 865-880 Request permission

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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