We provide information on a non trivial structure of phase space of the cubicView original: http://arxiv.org/abs/1202.2890
NLS on a three-edge star graph. We prove that, contrarily to the case of the
standard NLS on the line, the energy associated to the cubic focusing
Schr\"odinger 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.