Point-like perturbed fractional Laplacians through shrinking potentials of finite range
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Date
2018-03
Journal Title
Journal ISSN
Volume Title
Publisher
SISSA
Abstract
We construct the rank-one, singular (point-like) perturbations of
the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrödinger 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 Schrödinger operators formed by a fractional Laplacian and a regular potential.
Description
Keywords
Fractional Laplacian, Singular perturbations of differential operators, Schrödinger operators with shrinking potentials, Zero-energy resonance