Count of eigenvalues in the generalized eigenvalue problem

Abstract

We address the count of isolated and embedded eigenvalues in a 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 spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin's Invariant Subspace theorem in an indefinite inner product space but it 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 form defined by the indefinite inner product is strictly positive on the subspace related to the absolutely continuous part of the spectrum of the generalized eigenvalue problem. Applications to solitons and vortices of the nonlinear Schrödinger equations and solitons of the Korteweg-De Vries equations are developed from the general theory.

Stability of some stationary solutions to the forced KdV equation with one or two bumps

“…Then, according to Chugunova and Pelinovsky [24], the following relationship between the number of eigenmodes of L and those of ∂ x L holds:…”

Periodic Traveling Waves in Diatomic Granular Chains

“…Because sin(θ) = sin(π − θ), every non-zero eigenvalue corresponding to θ k (m, N ) = π 2 is double. Because all eigenvalues λ ∈ iR + have a definite Krein signature by Lemma 3 and the sign of σ (1) n in (68) is same for both eigenvalues with sin(θ) = sin(π− θ), the double eigenvalues λ ∈ iR are structurally stable with respect to parameter continuations [4] in the sense that they split along the imaginary axis beyond the leading-order perturbation theory.…”

“…for which the system of two granular chains (4) reduces to the scalar granular chain (a so-called monomer),Ü…”

Instability indices for matrix polynomials

“…Other referential works which study spectral problems with this perspective include Azizov and Iokhvidov [4], Gohberg et al [9], Shkalikov [34], and Shkalikov [35]. A different approach to studying the problem, based upon the techniques of spectral decomposition and simultaneous diagonalization of quadratic forms, is found in Chugunova and Pelinovsky [7], Hǎrǎguş and Kapitula [14], and Pelinovsky [32].…”