  We use the density matrix renormalization group (DMRG) to map out the ground
state of a XY-spin chain coupled to dispersionless phonons of frequency $%
\omega $. We confirm the existence of a critical spin-phonon coupling $% \alpha
_c\propto \omega ^{0.7}$ for the onset of the spin gap bearing the signature of
a Kosterlitz-Thouless transition. We also observe a classical-quantum crossover
when the spin-Peierls gap $\Delta $ is of order $% \omega $. In the classical
regime, $\Delta >\omega $, the mean-field parameters are strongly renormalized
by non-adiabatic corrections. This is the first application of the DMRG to
phonons.
