Lp-independence of spectral bounds of Schrodinger-type operators with non-local potentials

We establish a necessary and sufficient condition for spectral bounds of a non-local Feynman-Kac semigroup being L-p-independent. This result is an extension of that in [24] to more general symmetric Markov processes; in [24], we only treated a symmetric stable process on R-d. 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 L-P-independence.