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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...
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| Format: | Dissertation |
| Język: | angielski |
| Wydane: |
Philipps-Universität Marburg
2015
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| Hasła przedmiotowe: | |
| Dostęp online: | PDF pełnotekstowe |
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| Streszczenie: | 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. |
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| DOI: | 10.17192/z2015.0418 |