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