Semiclassical analysis and symmetry reduction I. Equivariant Weyl law for invariant Schr\"odinger operators on compact manifolds

Benjamin Küster; Pablo Ramacher

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arXiv · 1508.03540

Semiclassical analysis and symmetry reduction I. Equivariant Weyl law for invariant Schr\"odinger operators on compact manifolds

Benjamin Küster·Pablo Ramacher

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Abstract

We study the spectral properties of Schr\"odinger operators on a compact connected Riemannian manifold $M$ without boundary 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 semiclassical functional calculus for $h$-dependent functions and relying on recent results on singular equivariant asymptotics. These results will be used to derive an equivariant quantum ergodicity theorem in Part II of this work. When $G$ is trivial, one recovers the classical results.

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BibTeX RIS

Benjamin Küster, Pablo Ramacher. 2015-08-14. Semiclassical analysis and symmetry reduction I. Equivariant Weyl law for invariant Schr\"odinger operators on compact manifolds. https://arxiv.org/abs/1508.03540

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