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We consider a continuous-time symmetric supercritical branching random walkon a multidimensional lattice with a finite set of the particle generationcentres, i.e. branching sources. The main object of study is the evolutionaryoperator for the mean number of particles both at an arbitrary point and on theentire lattice. The existence of positive eigenvalues in the spectrum of anevolutionary operator results in an exponential growth of the number ofparticles in branching random walks, called supercritical in the such case. Forsupercritical branching random walks, it is shown that the amount of positiveeigenvalues of the evolutionary operator, counting their multiplicity, does notexceed the amount of branching sources on the lattice, while the maximal ofthese eigenvalues is always simple. We demonstrate that the appearance ofmultiple lower eigenvalues in the spectrum of the evolutionary operator can becaused by a kind of `symmetry' in the spatial configuration of branchingsources. The presented results are based on Green's function representation oftransition probabilities of an underlying random walk and cover not only thecase of the finite variance of jumps but also a less studied case of infinitevariance of jumps.

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails