  We study the relation between spectral invariants of disjointly supported
Hamiltonians and of their sum. On aspherical manifolds, such a relation was
established by Humili\`ere, Le Roux and Seyfaddini. We show that a weaker
statement holds in a wider setting, and derive applications to Polterovich's
Poisson bracket invariant and to Entov and Polterovich's notion of superheavy
sets.
