Exponential growth of the numbers of particles for branching symmetric α-stable processes

We study the exponential growth of the numbers of particles for a branching symmetric alpha-stable process in terms of the principal eigenvalue of an associated Schrodinger operator. Here the branching rate and the branching mechanism can be state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the Lebesgue measure. We calculate the principal eigenvalues and give some examples.