POTENTIAL THEORY OF SCHRÖDINGER OPERATOR BASED ON
FRACTIONAL LAPLACIAN
Krzysztof Bogdan
Tomasz Byczkowski
Abstract: 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 -harmonic functions. Results are obtained by analyzing properties of a
symmetric -stable Lévy process on 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.
1991 AMS Mathematics Subject Classification: Primary 31B2S, 60JS0.
Key words and phrases: symmetric -stable Lévy process, Feynman-Kac semigroup,
Schrödinger operator, -harmonic functions, Kelvin transform, conditional gauge theorem.