V. Vougalter and D. Pelinovsky

Eigenvalues of zero energy in the linearized NLS problem

Journal of Mathematical Physics 47, 062701 (2006)

Abstract:
We study a pair of neutrally stable eigenvalues of zero energy in the linearized NLS equation. We prove that the pair of isolated eigenvalues of geometric multiplicity two and algebraic multiplicity 2N is associated with 2 P negative eigenvalues of the energy operator, where P = N/2 if N is even and P = (N-1)/2 or P = (N+1)/2 if N is odd. When the potential of the linearized NLS problem is perturbed with a parameter continuation, we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy.

Keywords:
SPECTRAL THEORY, NONLINEAR SCHRODINGER EQUATION, LINEARIZED ENERGY, UNSTABLE EIGENVALUES, HAMILTON-HOPF BIFURCATION