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Equidistribution of phase shifts in semiclassical potential scattering

Consider the Hamiltonian H:=h2Δ+V–E where Δ is the positive Laplacian on Rd , V∈C0∞(Rd) is a smooth, compactly supported potential, E>0 is an energy level, and h>0 is a semiclassical parameter. We study the eigenvalues of the scattering matrix Sh(E) , which lie on the unit circle S1⊂C due to the unitarity of Sh(E) . Under an appropriate hypothesis on the classical dynamical flow corresponding to H , we show that in the limit h→0 , the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1∈S1 .