  We consider a class of non-local reaction-diffusion problems, referred to as
replicator-mutator equations in evolutionary genetics. For a confining fitness
function, we prove well-posedness and write the solution explicitly, via some
underlying Schr\"odinger spectral elements (for which we provide new and
non-standard estimates). As a consequence, the long time behaviour is
determined by the principal eigenfunction or ground state. Based on this, we
discuss (rigorously and via numerical explorations) the conditions on the
fitness function and the mutation rate for evolutionary branching to occur.
