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 two such 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 Schrödinger equations for the corresponding operators, we outline a programme of relevant questions that deserve being investigated.

VL - 59 UR - https://doi.org/10.1063/1.5033856 ER - TY - JOUR T1 - On fractional powers of singular perturbations of the Laplacian JF - Journal of Functional Analysis Y1 - 2018 A1 - Vladimir Georgiev A1 - Alessandro Michelangeli A1 - Raffaele Scandone KW - Point interactions KW - Regular and singular component of a point-interaction operator KW - Singular perturbations of the Laplacian AB -We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.

VL - 275 UR - http://www.sciencedirect.com/science/article/pii/S0022123618301046 ER - TY - RPRT T1 - Friedrichs systems in a Hilbert space framework: solvability and multiplicity Y1 - 2017 A1 - Nenad Antonić A1 - Marko Erceg A1 - Alessandro Michelangeli AB - The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide suffcient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples. UR - http://preprints.sissa.it/handle/1963/35280 U1 - 35587 U2 - Mathematics U4 - 1 ER -