Airy and Schr\"odinger-type equations on looping-edge graphs and applications

The aim of this work is to study the Airy and Schr\"odinger operators on looping-edge graphs, a class of metric graphs consisting of a circle and a finite number $N$ of infinite half-lines attached to a common vertex. For the Airy operator, we characterize all extensions generating unitary and contractive dynamics in terms of self-orthogonal subspaces and linear operators acting on indefinite inner product spaces (Krein spaces) associated to the boundary values at the vertex. Employing similar abstract techniques, we then describe a systematic way to produce self-adjoint extensions of the Schr\"odinger operator that are compatible with prescribed boundary relations on looping-edge and $\mathcal{T}$-shaped graphs.