We consider the scattering of particles (kinetic energy eps) by an obstacle which tunnels coherently between two positions (tunnel splitting Delta), for arbitrary values of eps/Delta and scattering strength U. The obstacle mimics two classical scatterers at fixed positions when eps lesseq Delta. Interference disappears when eps >> Delta, but can be recovered if the elastic-scattering channel is detected only. At intermediate values of eps there is a systematic interplay of the coherent tunnel motion and the ballistic particle motion. We show that the transmission probability can remain finite even in the limit U->infinity because the particle can evade the obstacle systematically. We discuss the realization of a quantum obstacle in mesoscopic systems
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