Spectra of positive and negative energies in the linearized NLS problem

By Scipio Cuccagna, Dmitry E. Pelinovsky, V. Vougalter. 2005

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We study the spectrum of the linearized NLS equation in three dimensions in association with the energy spectrum. We prove that unstable eigenvalues of the linearized NLS problem are related to negative eigenvalues of the energy spectrum, while neutrally stable eigenvalues may have both positive and negative energies. The nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum. We derive bounds on the number of unstable eigenvalues of the linearized NLS problem and study bifurcations of embedded eigenvalues of positive and negative energies. We develop the L2‐scattering theory for the linearized NLS operators and recover results of Grillakis [5] with a Fermi golden rule. © 2004 Wiley Periodicals, Inc.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/db52e04518214da55f4992b423f2cea62a176e94

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Citations in the corpus

Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems (2004, unknown): “∗E-mail: kapitula@math.unm.edu †E-mail: kevrekid@math.umass.edu ‡E-mail: sandstede.1@osu.edu”
Count of eigenvalues in the generalized eigenvalue problem (2006, methodology): “The rst result is a remake of the main results obtained in [ CPV05 , KKS04, P05], although the method of proof presented therein is quite dierent than that given here.”
Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs (2017, background): “Like in the literature ([44] [21]), the index formula was usually proved by reducing the eigenvalue problem JLu = λu to a generalized eigenvalue problem (R− zS) v = 0 (so called linear operator pencil), where z = −λ2 and R, S are self-adjoint operators with ker S = {0}.”
On the spectra of periodic waves for infinite-dimensional Hamiltonian systems (2008, background): “The main purpose of this paper is to obtain a general result relating the spectra of the operators L and JL, in the spirit of the results in [6, 18, 19, 25] for localized waves, in which the total number of negative point eigenvalues of L is related to the total number of unstable eigenvalues of JL.”
Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems (2012, background): “(1.19) Note that the sign under the exponent here is negative, which corresponds to the case of so-called embedded solitons (see e.g. [9]), as distinct from the standard situation (see e.g. [2], [12]).”