  Phonon cavities are believed to be the next step towards a control of
dephasing in semiconductor quantum dot `qubits'. In this paper, we discuss two
models for phonon cavities - a surface acoustic wave (SAW) inter-digitated
transducer on an infinite half-space, and an elastic thin slab. The inelastic
current through double quantum dots in non-perfect SAW cavities exhibits a gap
at small energies and is completely suppressed in a perfect, infinite system.
In the free-standing slab model, van Hove singularities evolve in the phonon
spectral density. We find that these singularities cause additional side peaks
in the inelastic current.
