S. Cuccagna, D. Pelinovsky, and V. Vougalter

Spectra of positive and negative energies in the linearized NLS problem

Comm. Pure Appl. Math. 58, 1-29 (2005)

Abstract:
We study the spectrum of the linearized NLS equation in three and higher 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 non-singular 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.

Keywords:
SPECTRAL THEORY, NONLINEAR SCHRODINGER EQUATION, EMBEDDED EIGENVALUES, END POINTS, RESONANCES, BIFURCATIONS OF EIGENVALUES AND RESONANCES, ENERGY FUNCTIONALS, STABILITY OF SOLITARY WAVES, WAVE OPERATORS