  Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter,
2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing
high-frequency Helmholtz solutions into "low"- and "high"-frequency components
have had a large impact in the numerical analysis of the Helmholtz equation.
These results have been proved for the constant-coefficient Helmholtz equation
in either the exterior of a Dirichlet obstacle or an interior domain with an
impedance boundary condition.
  Using the Helffer-Sj\"ostrand functional calculus, this paper proves
analogous decompositions for scattering problems fitting into the black-box
scattering framework of Sj\"ostrand-Zworski, thus covering Helmholtz problems
with variable coefficients, impenetrable obstacles, and penetrable obstacles
all at once.
  These results allow us to prove new frequency-explicit convergence results
for (i) the $hp$-finite-element method applied to the variable coefficient
Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle
and coefficients are analytic, and (ii) the $h$-finite-element method applied
to the Helmholtz penetrable-obstacle transmission problem. In particular, the
result in (i) shows that the $hp$-FEM applied to this problem does not suffer
from the pollution effect.

Analysis of PDEs;Numerical Analysis;Numerical Analysis