L^p-independence of spectral bounds of Schrödinger type semigroups

Let L be an m-symmetric Markov generator and μ a signed measure in the Kato class. We consider a Schrödinger type operator H^μ = - L + μ on L^p (m). We prove that under certain conditions for the Markov semigroup generated by L and the potential μ, the L^p-spectral bound of H^μ is independent of p if and only if the L²-spectral bound is non-positive.