Semiclassical Analysis of Schrödinger Operators on Closed Manifolds and Symmetry Reduction

Let M be a closed connected Riemannian manifold. In the first part of this thesis, we develop a functional calculus for h-dependent functions within the theory of semiclassical pseudodifferential operators. Our results lead to semiclassical trace formulas with remainder estimates that are well-suite...

Mô tả đầy đủ

Đã lưu trong:
Chi tiết về thư mục
Tác giả chính: Küster, Benjamin
Tác giả khác: Ramacher, Pablo (Prof. Dr.) (Cố vấn luận án)
Định dạng: Dissertation
Ngôn ngữ:Tiếng Anh
Được phát hành: Philipps-Universität Marburg 2015
Những chủ đề:
Truy cập trực tuyến:Bài toàn văn PDF
Các nhãn: Thêm thẻ
Không có thẻ, Là người đầu tiên thẻ bản ghi này!
Miêu tả
Tóm tắt:Let M be a closed connected Riemannian manifold. In the first part of this thesis, we develop a functional calculus for h-dependent functions within the theory of semiclassical pseudodifferential operators. Our results lead to semiclassical trace formulas with remainder estimates that are well-suited for studying spectral windows of width of order h^d, where 0 < d < 1/2. In the second part of the thesis, we study the spectral and quantum ergodic properties of Schrödinger operators on M in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, if M carries an isometric and effective action of a compact connected Lie group G, we prove a generalized equivariant version of the semiclassical Weyl law with an estimate for the remainder, using a theorem from the first part of this thesis and relying on recent results on singular equivariant asymptotics. We then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of M is ergodic. In particular, we obtain an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdiere theorem, as well as a representation theoretic equidistribution theorem. If M/G is an orbifold, similar results were recently obtained by Kordyukov. When G is trivial, one recovers the classical results.
DOI:10.17192/z2015.0418