Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology

Abstract

Abstract. We review some recent results on the minimization of the energy associated to the nonlinear Schrödinger Equation on non-compact graphs. Starting from seminal results given by the author together with C. Cacciapuoti, D. Finco, and D. Noja for the star graphs, we illustrate the achiements attained for general graphs and the related methods, developed in collaboration with E. Serra and P. Tilli. We emphasize ideas and examples rather than computations or proofs.

Index theorems for graph-parametrized optimal control problems

“…and it is identified with functions which are locally constant on [−1, 0] and on [1,2] with values in R dim N0 and R dim N1 respectively. In Figure 5 the construction is explained visually.…”

“…Since the vector field associated to ũ is zero on [−1, 0] ∪ [1,2] the flow is a constant transformation. Composing the Hamiltonian with the flow Φt gives us bt û(λ) = ĥû − ĥũ(t)…”

“…Then we can define Ẑt = ∂ û− → b t û| λ=λ(0) . and denote Z 0 and Z 1 to be restrictions of Ẑt to the time intervals [−1, 0] and [1,2] correspondingly, so that…”

“…There is an extensive literature concerning quantum graphs. In particular regarding existence of global minimizers for the NLS energy, see for example [4], [5], [3], [17] (or [15] and [2] for overviews of the results) and references therein. Also in situation that do not fit in our current framework, for example [22] where the same problem is discussed on a graph with an infinite number of edges.…”

Standing waves on quantum graphs

“…The linear evolutionary models and the spectral theory for quantum graphs are well covered by several monographs [16,31,60] and many recent publications. The first review of the nonlinear evolutionary models was published some time ago [98] and was complemented by the recent reviews [1,11,56] which explained various technical aspects of mathematical analysis of the ground state on the quantum graphs. Compared to these publications, we would like to present a general overview of how the standing waves arise in the nonlinear models and how their existence and stability can be analyzed with different analytical methods.…”