Mass-constrained ground states of the stationary NLSE on periodic metric graphs

Abstract

We investigate the existence of ground states with fixed mass for the nonlinear Schrödinger equation with a pure power nonlinearity on periodic metric graphs. Within a variational framework, both the L 2 -subcritical and critical regimes are studied. In the former case, we establish the existence of global minimizers of the NLS energy for every mass and every periodic graph. In the critical regime, a complete topological characterization is derived, providing conditions which allow or prevent ground states of a certain mass from existing. Besides, a rigorous notion of periodic graph is introduced and discussed.

Quantum graphs and dimensional crossover: the honeycomb

“…The analysis of NLS equations on periodic graphs has been developed for instance in ( [16,17,18]), and a systematic discussion of the problem of ground states for periodic graphs has been carried out in [21], however here we shall focus on a particular phenomenon highlited in [4] and called dimensional crossover. Investigating the problem of proving the existence or the nonexistence of ground states for the NLS on the regular two-dimensional square grid (see Figure 1), it was found that three different regimes come into play:…”

NLS ground states on metric trees: existence results and open questions

“…Similarly, using (17) instead of (18), one obtains (27). Finally, (26) is obtained combining (28) with the inequality…”

“…More recently, however, another interesting class of graphs has been investigated, where noncompactness is no longer due to the presence of unbounded edges, but rather to the infinite number of bounded edges, arranged to form an infinite periodic structure [3, 4, 17, 28]. A prototypical case study can be found in [4], where the graph is a planar grid with vertices on the lattice , connected by vertical and horizontal edges of length one: the main interesting feature is that, even though such an ambient space is of course 1‐dimensional (at least locally), at large scales the overall behavior turns out to be 2‐dimensional, to the extent that a Sobolev inequality holds true, formally identical to the Sobolev inequality in .…”

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

“…For the sake of completeness, we also remark that the NLSE on compact graphs (which, in particular, do not fulfill (H1)) has been studied, e.g., in [20,24,28,38]; while the case of one or higher-dimensional periodic graphs (which, in particular, do not fulfill (H2) as, for instance, in Figure 3) has been addressed, e.g., by [6,25,32,43].…”