Bifurcations from the end points of the essential spectrum in the linearized NLS problem

We study bifurcations of eigenvalues from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem in three dimensions. We show that a resonance and an eigenvalue of positive energy at the endpoint may bifurcate only to a real eigenvalue of positive energy, while an eigenvalue of negative energy at the endpoint may also bifurcate to complex eigenvalues.