1998
DOI: 10.1103/physreve.57.5425
|View full text |Cite|
|
Sign up to set email alerts
|

Rate of quantum ergodicity in Euclidean billiards

Abstract: Abstract:For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdière and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of e… Show more

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1

Citation Types

0
86
0

Year Published

2000
2000
2013
2013

Publication Types

Select...
5
2

Relationship

0
7

Authors

Journals

citations
Cited by 73 publications
(86 citation statements)
references
References 58 publications
0
86
0
Order By: Relevance
“…For bounded systems (µ = λ) and for ǫ = 0 we obtain the expression for the variance of matrix elements derived in [13] and tested and verified in many situations [14][15][16]. If the system is sufficiently open so that ΓT H ≫ 1 the contributions from the off-diagonal terms can be neglected and the integration continued to infinity (as explained in the next paragraph).…”
Section: Semiclassical Correlation Functions For Smooth Operatorsmentioning
confidence: 99%
“…For bounded systems (µ = λ) and for ǫ = 0 we obtain the expression for the variance of matrix elements derived in [13] and tested and verified in many situations [14][15][16]. If the system is sufficiently open so that ΓT H ≫ 1 the contributions from the off-diagonal terms can be neglected and the integration continued to infinity (as explained in the next paragraph).…”
Section: Semiclassical Correlation Functions For Smooth Operatorsmentioning
confidence: 99%
“…1b which falls one side of the straight line ∂Ω A \Γ. Our choice of the shape of Ω A was informed by the issue of boundary effects raised by Bäcker et al [5], the main point being that within a boundary layer of order a wavelength, there are Gibbs-type phenomena associated with spectral projections, and by choosing a large angle of intersection of the line with Γ their contribution is minimized. Our classical mean is A = vol(Ω A )/ vol(Ω) ≈ 0.55000.…”
Section: Computing Quantum Matrix Elementsmentioning
confidence: 99%
“…1). Numerical tests of Conjecture 1 (in the form of S 1 (E; A), see Remark 1.1) exist for low eigenvalues (n < 6000) in the Anosov cardioid billiard [5]: various powers were found in the range γ = 0.37 to 0.5, and up to 20% deviations from the prefactor in Conjecture 2.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…For integrable systems these are the invariant tori. For ergodic dynamics the eigenstates become equidistributed on the energy shell [2]. Typical systems have a mixed phase space, where regular islands and chaotic regions coexist.…”
mentioning
confidence: 99%