  We present an approach to calculating the quantum resonances and resonance
wave functions of chaotic scattering systems, based on the construction of
states localized on classical periodic orbits and adapted to the dynamics.
Typically only a few of such states are necessary for constructing a resonance.
Using only short orbits (with periods up to the Ehrenfest time), we obtain
approximations to the longest living states, avoiding computation of the
background of short living states. This makes our approach considerably more
efficient than previous ones. The number of long lived states produced within
our formulation is in agreement with the fractal Weyl law conjectured recently
in this setting. We confirm the accuracy of the approximations using the open
quantum baker map as an example.
