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M. Chugunova and D. Pelinovsky

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Count of eigenvalues in the generalized eigenvalue problem

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Journal of Mathematical Physics 51, 052901-19 (2010)

**Abstract:**

We study isolated and embedded eigenvalues in the generalized eigenvalue problem
defined by two self-adjoint operators with a positive essential spectrum and a
finite number of isolated eigenvalues. The generalized eigenvalue problem determines
the spectral stability of nonlinear waves in infinite-dimensional Hamiltonian
systems. The theory is based on Pontryagin’s invariant subspace theorem and extends
beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others.
Our main results are (i) the number of unstable and potentially unstable eigenvalues
equals the number of negative eigenvalues of the self-adjoint operators, (ii) the
total number of isolated eigenvalues of the generalized eigenvalue problem is
bounded from above by the total number of isolated eigenvalues of the self-adjoint
operators, and (iii) the quadratic forms defined by the two self-adjoint operators are
strictly positive on the subspace related to the continuous spectrum of the generalized
eigenvalue problem. Applications to the localized solutions of the nonlinear
Schrödinger equations are developed from the general theory.

**Keywords**:

Generalized eigenvalue problem, Discrete and continuous spectrum,
Indefinite metric, Invariant subspaces, Krein signature, Nonlinear Schrodinger equation,
Solitary waves, Vortices