Branching random walks on $\mathbb{Z}$ with one particle generation center and symmetrically located absorbing sources

We consider a time-continuous branching random walk on a one-dimensional lattice on which there is one center (lattice point) of particle generation, called branching source. The generation of particles in the branching source is described by a Markov branching process. Some number (finite or infinite, depending on the problem formulation) of absorbing sources is located symmetrically around the branching source. For such configurations of sources we obtain necessary and sufficient conditions for the existence of an isolated positive eigenvalue of the evolution operator. It is shown that, under some additional assumptions, the existence of such an eigenvalue leads to an exponential growth of the number of particles in each point of the lattice.