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On the Schroedinger operator based on the fractional Laplacian

Bogdan K., Byczkowski T.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 3

291-301

We announce new results in the potential theory of Schroedinger operators based on the fractional Laplacian on Euclidean spaces of arbitrary dimension. We concentrate on questions related to gaugeability and existence of q-harmonic functions. Results are obtained by analyzing properties of symmetric [alpha]-stable Levy processes on Rd, including the recurrent case. We also provide some explicit examples of gauge functions for a general class of domains.

Potential theory of Schrödinger operator based on fractional Laplacian

Bogdan K., Byczkowski T.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

293--335

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.