Eigenvalues of zero energy in the linearized NLS problem

We study a pair of neutrally stable eigenvalues of zero energy in the linearized NLS equation. We prove that the pair of isolated eigenvalues, where each eigenvalue has geometric multiplicity one and algebraic multiplicity N, is associated with 2P 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 due to parameter continuations, we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy.