Semiclassical Analysis of Schrödinger Operators on Closed Manifolds and Symmetry Reduction
Semiklassische Analysis von Schrödingeroperatoren auf geschlossenen Riemannschen Mannigfaltigkeiten und Symmetriereduktion
Benjamin
Küster
Mathematics
Mathematik
Geometrische Analysis
spectral theory
Schrödinger operator
Spektraltheorie
Schrödinger Operator
Geometric Analysis
singular symplectic reduction
singuläre symplektische Reduktion
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.
Philipps-Universität Marburg
Marburg
Deutschhausstraße 9, 35037 Marburg
Pablo
Ramacher
Prof. Dr.
2015
2015-12-04
2015-12-23
Text
doctoralThesis
urn:nbn:de:hebis:04-z2015-04188
eng
thesis.doctoral
Philipps-Universität Marburg
Marburg
1
https://archiv.ub.uni-marburg.de/diss/z2015/0418/pdf/dbk.pdf
10.17192/z2015.0418
https://archiv.ub.uni-marburg.de/diss/z2015/0418
https://creativecommons.org/licenses/by-nc-nd/4.0/