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Consider the equation ( * ) ½Δu-Vu=0 in R d , for d > 3, where V ≥ 0. One says that the Liouville property holds if the only bounded solution to ( * ) is 0. Under a certain growth condition on V, we give a necessary and sufficient analytic condition for the Liouville property to hold. We then apply this to the case that V is the indicator function of an infinite collection of small balls.

A probabilistic approach to the Liouville property for Schrödinger operators with an application to infinite configurations of balls