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V. Vougalter and D. Pelinovsky

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Eigenvalues of zero energy in the linearized NLS problem

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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