Stability theory for the NLS equation on looping edge graphs

The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on looping edge graphs, namely, a graph consisting of a circle with several half-lines attached at a single vertex. The main novelty of this paper is at least twofold: by considering \(\delta \)-type boundary conditions at the vertex, the extension theory of Krein &von Neumann, and a splitting eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around of a priori positive single-lobe state profile for every positive power, this information will be main for a local stability study; and so via a bifurcation analysis on the phase plane we build at least two families of positive single-lobe states and we study the stability properties of these in the subcritical, critical, and supercritical cases. Our results recover some spectral studies in the literature associated to the NLS on looping edge graphs which were obtained via variational techniques.