  We study the local classical and quantum critical properties of
electron-vibration interaction, represented by the Yu-Anderson model. It
exhibits an instability, similar to the Wentzel-Bardeen singularity, whose
nature resembles to weakly first order quantum phase transitions at low
temperatures, and crosses over to Gaussian behaviour with increasing
temperature. We determine the dominant energy scale separating the quantum from
classical criticality, study the effect of dissipation and analyze its impact
on correlation functions. Similar phenomenon should be observable in carbon
nanotubes around local defects.
