On the lack of bound states for certain NLS equations on metric graphs

Abstract

The purpose of this paper is to prove some results on the absence of bound states for certain nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearity. In particular, we show how the topological and metric properties of graphs affect the existence/nonexistence of bound states. This work completes the discussion initiated in [17, 18].Comment: 20 pages, 9 figure

Existence of infinitely many stationary solutions of the L2-subcritical and critical NLSE on compact metric graphs

“…Moreover, existence of ground states and bound states on non-compact graphs was investigated also for the NLS equation with concentrated nonlinearity in [23,22,21]. Specifically, the general scheme followed in [21] provides the tools we will use in this paper when dealing with bound states (see Section 2).…”

Standing waves on quantum graphs

“…The problem was addressed for some metric graphs in [68], where the existence of many bound states in the presence of a rather complex pattern of scattering resonances was shown by means of numerical approximations. Analysis of ground states and more generally bound states for a metric graph with localized nonlinearity has been treated in the papers [111,112,115] in the subcritical case 0 < p < 2 and in [58] in the critical case p = 2.…”

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

“…Within this framework, there has been an intensive study of the existence of mass-constrained ground states for the NLS energy, that is, global minimizers of the energy among functions of prescribed L 2 norm. This problem has been initially considered in the case of graphs made up of a core of finitely many bounded edges, and a finite number of unbounded edges (half lines) attached to it, and this setting is nowadays quite well understood (we refer to [5][6][7] for the nonlinearity extended to the whole graph, and to [18,19,30,31,34] for the nonlinearity concentrated on the sole compact core). Similar results have then been accomplished also in the case of compact graphs [13,16,24].…”