منابع مشابه
Note on Quantum Unique Ergodicity
The purpose of this note is to record an observation about quantum unique ergodicity (QUE) which is relevant to the recent construction of H. Donnelly [D] of quasi-modes on nonpositively curved surfaces and to similar examples known as bouncing ball modes [BSS, H] on stadia. It gives a rigorous proof of a localization statement of Heller-O’Connor [HO] for eigenfunctions of the stadium. The rele...
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LetM be a complete Riemannian manifold with finite volume which we initially assume to be compact. Then since M is compact, L(M) is spanned by the eigenfunctions of the Laplacian ∆ on M . Many interesting questions can be asked about these eigenfunctions and their properties, and of these we focus on one, quantum ergodicity, which to the best of my knowledge was first considered by A. I. Šnirel...
متن کاملInvariant Measures and Arithmetic Quantum Unique Ergodicity
We classify measures on the locally homogeneous space Γ\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 can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In th...
متن کاملQuantum Unique Ergodicity and Number Theory
A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic surfacess arising as a quotient of the upper half-plane by a discrete “arithmetic” subgroup of SL2(R) (for example, SL2(Z), and in this case the corr...
متن کاملQuantum Unique Ergodicity for Parabolic Maps
We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical phase-space average with respect to Liouville measure of the corresponding classical observable. The possible existence of any exceptional subsequences of eigenstates is an i...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2002
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-02-06810-7