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

TL;DR

This paper investigates the energy landscape of the cubic focusing NLS on a three-edge star graph, revealing unique critical point structures that differ from the standard line case, including the absence of energy minima and the saddle nature of stationary states.

Contribution

It characterizes the critical energy levels and stationary states of the cubic focusing NLS on a star graph, highlighting differences from classical NLS on the line.

Findings

Energy does not attain a minimum on any sphere of fixed L^2-norm.

The only stationary state with prescribed L^2-norm is a saddle point.

The structure of phase space differs significantly from the standard NLS case.

Abstract

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.