Error of semiclassical eigenvalues in the semiclassical limit - An asymptotic analysis of the Sinai billiard

P. Dahlqvist

doi:10.1088/0305-4470/32/42/307

Home/Archive/arXiv

arXiv · chao-dyn/9812017

Error of semiclassical eigenvalues in the semiclassical limit - An asymptotic analysis of the Sinai billiard

P. Dahlqvist

View original source ↗Read source document ↗Export metadata ↓

Abstract

We estimate the error in the semiclassical trace formula for the Sinai billiard under the assumption that the largest source of error is due to Penumbra diffraction, that is diffraction effects for trajectories passing within a distance R O((kR)^(-2/3) to the disk and trajectories being scattered in very forward directions. Here k is the momentum and R the radius of the scatterer. The semiclassical error is estimated by perturbing the Berry-Keating formula. The analysis necessitates an asymptotic analysis of very long periodic orbits. This is obtained within an approximation originally due to Baladi Eckmann and Ruelle. We find that the average error, for sufficiently large value of kR, will exceed the mean level spacing.

Explore related subjects

chao-dyn nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

P. Dahlqvist. 1998-12-09. Error of semiclassical eigenvalues in the semiclassical limit - An asymptotic analysis of the Sinai billiard. https://doi.org/10.1088/0305-4470%2F32%2F42%2F307

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2024

Dynamical ensembles equivalence in fluid mechanics

Dissipative Euler and Navier Stokes equations are discussed with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis.

chao-dyn↗

arXiv2015

The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System

We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Hénon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the manifolds of a remote saddle point. The Bogdanov map is the Euler map of a two-dimensional system of ordinary differential equations first considered by Bogdanov and Arnold in their study of the versal unfolding of the double-zero-eigenvalue singularity, and equivalently of a vector field invariant under rotation of the plane by an angle $2π$. It is a useful system in which to observe the effect of dissipative perturbations on Hamiltonian structure. In addition, we argue that the Bogdanov map provides a good approximation to the dynamics of the Poincaré maps of periodically forced oscillators.

chao-dyn↗

arXiv2012

Passive Scalars and Three-Dimensional Liouvillian Maps

Global aspects of the motion of passive scalars in time-dependent incompressible fluid flows are well described by volume-preserving (Liouvillian) three-dimensional maps. In this paper the possible invariant structures in Liouvillian maps and the two most interesting nearly-integrable cases are investigated. In addition, the fundamental role of invariant lines in organizing the dynamics of this type of system is exposed. Bifurcations involving the destruction of some invariant lines and tubes and the creation of new ones are described in detail.

chao-dyn↗