We construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional SchrÃ¶dinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for SchrÃ¶dinger operators formed by a fractional Laplacian and a regular potential.

UR - https://doi.org/10.1007/s11785-019-00927-w ER - TY - RPRT T1 - On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians Y1 - 2016 A1 - Alessandro Michelangeli A1 - Andrea Ottolini AB - For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the ``Ter-Martirosyan--Skornyakov condition'' gives rise to a self-adjoint realisation of the corresponding Hamiltonian. This is done within the self-adjoint extension scheme of Krein, Visik, and Birman. We show that the Ter-Martirosyan--Skornyakov asymptotics is a condition of self-adjointness only when is imposed in suitable functional spaces, and not just as a point-wise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature. UR - http://urania.sissa.it/xmlui/handle/1963/35195 U1 - 35489 U2 - Mathematics U4 - 1 U5 - MAT/07 ER -