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

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

2012-02-13

arxiv.org/abs/1202.2890v1

Physics

Mathematical Physics

Scientific paper

We provide information on a non trivial structure of phase space of the cubic 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.

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