On the structure of critical energy levels for the cubic focusing NLS on star graphs

We provide information on a non-trivial structure of phase space of the cubic nonlinear Schrodinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrödinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L-2-norm. We moreover show that the only stationary state with prescribed L-2-norm is indeed a saddle point.