2009/09/11 by Joachim Hilgert, Hilgert, Joachim, Michael Schroeder +1
Mathematics · Physics and Astronomy · #53C35 #58C40 #58J50 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:53C35 #msc:58C40 #msc:58J50
paper · pdf · doi:10.48550/arxiv.0909.2142
25 pages
arxiv created 2009/09/18 · arxiv updated 2009/12/01
There is a remarkable relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold: It was observed in \citeAZ that for compact hyperbolic surfaces XΓ=Γ\backslashℍ Wigner distributions ∫S^* XΓ a dWirj = < Op(a)ϕirj,ϕirj >L2(XΓ) and Patterson--Sullivan distributions PSirj are asymptotically equivalent as rj→∞. We generalize the definitions of these distributions to all rank one symmetric spaces of noncompact type and introduce off-diagonal elements PSλj,λk. Further, we give explicit relations between off-diagonal Patterson--Sullivan distributions and off-diagonal Wigner distributions and describe the asymptotic relation between these distributions.