2014/10/04 by Benjamin Küster, Küster, Benjamin, Pablo Ramacher +1 · 2 citations
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1410.1096
This preprint has been withdrawn and superseded by the two preprints arXiv:1508.03540 and arXiv:1508.07381
arxiv created 2016/02/12 · arxiv updated 2016/02/15
We study the ergodic properties of eigenfunctions of Schrödinger operators on a closed connected Riemannian manifold M in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let M carry an isometric effective action of a compact connected Lie group G. We prove an equivariant quantum ergodicity theorem assuming that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of M is ergodic. We deduce the theorem by proving an equivariant version of the semiclassical Weyl law, relying on recent results on singular equivariant asymptotics. It implies an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdière theorem, as well as a representation theoretic equidistribution theorem. In case that G is trivial, one recovers the classical results.