Fractional powers and singular perturbations of quantum differential Hamiltonians

We consider the fractional powers of singular (point-like) perturbations of the Laplacian, and the singular perturbations of fractional powers of the Laplacian, and we compare such two constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schr\"{o}dinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated.