  We study the Helmholtz equation for a heterogeneous system in $d$ dimensions
and show that it is possible to calculate exactly the sum rules of rational
order using perturbation theory by relating the sum rules to suitable traces.
The present calculation is restricted to a positive definite spectrum, e.g.
Dirichlet boundary conditions, and it reproduces the result previously obtained
in Ref.~\cite{Amore12}, working directly with Rayleigh-Schr\"odinger
perturbation theory.
