We consider the Schr¨odinger operator on the unit circle, whose potential is an
Ornstein – Uhlenbeck type process, with drift depending on its position. We describe
the distribution of the periodic groundstate, based on the circular brownian motion
measure. The results exposed here, have been mentioned, but not proved, in [7].
Keywords: Schr¨odinger Operator, Ornstein – Uhlenbeck Process, Periodic Groundstate,
Circular Brownian Motion.