Semiclassical analysis and symmetry reduction I. Equivariant Weyl law for invariant Schrödinger operators on compact manifolds

2015/08/14 by Benjamin Küster, Küster, Benjamin, Pablo Ramacher +1
Mathematics · #58C40 #58J50 #58J51 #81Q10 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Geometry and complex manifolds #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1508.03540

openalex publication_date 2015/08/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the spectral properties of Schrödinger 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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