2013/11/11 by Jesse Gell-Redman, Andrew Hassell, Steve Zelditch
Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Equidistributed sequence #Geometry and complex manifolds #Hamiltonian (control theory) #Limit (mathematics) #Limit point #Quantum chaos and dynamical systems #Scattering #Semiclassical physics #Spectral Theory in Mathematical Physics #Unit circle #Unitarity #math.SP
paper · pdf · doi:10.1112/jlms/jdu068
published as J. London Math. Soc. (1) 91 (2015), 159-179 · 18 pages, 1 figure
arxiv created 2013/11/11 · openalex publication_date 2015/01/16 · arxiv updated 2015/06/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.