We establish a necessary and sufficient condition for spectral bounds of a non-local Feynman-Kac semigroup being Lp-independent. This result is an extension of that in  to more general symmetric Markov processes; in , we only treated a symmetric stable process on Rd. For example, we consider a symmetric stable process on the hyperbolic space, the jump process generated by the fractional power of the Laplace-Beltrami operator, and prove that by adding a non-local potential, the associated Feynman-Kac semigroup satisfies the Lp-independence.
Yoshihiro TAWARA. "Lp-independence of spectral bounds of Schrödinger-type operators with non-local potentials." J. Math. Soc. Japan 62 (3) 767 - 788, July, 2010. https://doi.org/10.2969/jmsj/06230767