# Semiclassical analysis and symmetry reduction II. Equivariant quantum ergodicity for invariant Schr\"odinger operators on compact manifolds

@article{Kuster2015SemiclassicalAA, title={Semiclassical analysis and symmetry reduction II. Equivariant quantum ergodicity for invariant Schr\"odinger operators on compact manifolds}, author={Benjamin Kuster and Pablo Ramacher}, journal={arXiv: Spectral Theory}, year={2015} }

We study the ergodic properties of Schrodinger operators on a compact connected Riemannian manifold $M$ without boundary in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an isometric and effective action of a compact connected Lie group $G$. Relying on an equivariant semiclassical Weyl law proved in Part I of this work, we deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on…

## 2 Citations

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

- Mathematics
- 2015

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…

### On the semiclassical functional calculus for h-dependent functions

- Mathematics
- 2015

We study the functional calculus for operators of the form $$f_h(P(h))$$fh(P(h)) within the theory of semiclassical pseudodifferential operators, where $$\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty…

## References

SHOWING 1-10 OF 46 REFERENCES

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

- Mathematics
- 2015

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…

### On Quantum Unique Ergodicity for Locally Symmetric Spaces

- Mathematics
- 2004

Abstract.We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a semi-canonical fashion, limits of eigenfunction measures on locally…

### Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

- Mathematics
- 2004

For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the…

### Classical Limits of Eigenfunctions for Some Completely Integrable Systems

- Mathematics
- 1999

We give an overview of some old results on weak* limits of eigenfunctions and prove some new ones. We first show that on M = (S n , can) every probability measure on S * M which is invariant under…

### Invariant measures and arithmetic quantum unique ergodicity

- Mathematics
- 2006

We classify measures on the locally homogeneous space ?i\ SL(2,R) ?~ L which are invariant and have positive entropy under the diagonal subgroup of SL(2,R) and recurrent under L. This classification…

### Stratified symplectic spaces and reduction

- Mathematics
- 1991

Let (M, w) be a Hamiltonian G-space with proper momentum map J: M -> g*. It is well-known that if zero is a regular value of J and G acts freely on the level set J '(0), then the reduced space MO =…

### Reduction and the trace formula

- Mathematics
- 1990

1. This is the second paper of a series in which we explore the relationships between the spectrum of an elliptic operator with symmetries and the procedure of reduction in classical mechanics. The…

### Scarred Eigenstates for Quantum Cat Maps of Minimal Periods

- Mathematics
- 2003

In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, shows a strong scarring phenomenon on the periodic orbits of the…

### Weyl's law and quantum ergodicity for maps with divided phase space

- Mathematics
- 2004

For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the…

### Ergodic billiards that are not quantum unique ergodic

- Mathematics
- 2008

Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in…