Count of eigenvalues in the generalized eigenvalue problem
TLDR
In this article, 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 eigen values were studied.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. Applicatio...read more
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References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
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Optical solitons : from fibers to photonic crystals
TL;DR: In this article, the authors introduce spatial and temporal solitons in photonic crystals, and introduce the concept of Incoherent Solitons, which is a subclass of the spatial and temporally soliton.
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Stability theory of solitary waves in the presence of symmetry, II☆
TL;DR: In this article, the effect of group invariance on the stability of solitary waves was studied and applications were given to bound states and traveling wave solutions of nonlinear wave equations, where the authors considered an abstract Hamiltonian system which is invariant under a group of operators.
Book
Oscillation matrices and kernels and small vibrations of mechanical systems
F. R. Gantmacher,M. Krein +1 more
TL;DR: A review of matrices and quadratic forms Oscillatory matrices Small oscillations of mechanical systems with $n$ degrees of freedom Small oscillational matrices with an infinite number of degree of freedom Sign-definite matrices A method of approximate calculation of eigenvalues and eigenvectors of an oscillatory matrix On a remarkable problem for a string with beads and continued fractions of Stieltjes Remarks References Index