  We introduce a novel multi-resolution Localized Orthogonal Decomposition
(LOD) for time-harmonic acoustic scattering problems that can be modeled by the
Helmholtz equation. The method merges the concepts of LOD and operator-adapted
wavelets (gamblets) and proves its applicability for a class of complex-valued,
non-hermitian and indefinite problems. It computes hierarchical bases that
block-diagonalize the Helmholtz operator and thereby decouples the
discretization scales. Sparsity is preserved by a novel localization strategy
that improves stability properties even in the elliptic case. We present a
rigorous stability and a-priori error analysis of the proposed method for
homogeneous media. In addition, we investigate the fast solvability of the
blocks by a standard iterative method. A sequence of numerical experiments
illustrates the sharpness of the theoretical findings and demonstrates the
applicability to scattering problems in heterogeneous media.
