We develop potential theory of Schrödinger operators based on fractional Laplacian on Euclidean spaces of arbitrary dimension. We focus on questions related to gaugeability and existence of q-harmonic functions. Results are obtained by analyzing properties of a symmetric α-stable Lévy process on Rd, including the recurrent case. We provide some relevant techniques and apply them to give explicit examples of gauge functions for a general class of domains.
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We prove the intrinsic ultracontractivity for the semigroup P[...] generated by the symmetric [alpha]-stable process killed on exiting a bounded open set D. The same property is valid for the alpha-stable Feynman-Kac semigroup T[...] provided D is a gaugeable bounded Lipschitz domain.
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