Random matrix elements and eigenfunctions in chaotic systems
Abstract
The expected root-mean-square value of a matrix element Aαβ in a classically chaotic system, where A is a smooth, ħ-independent function of the coordinates and momenta, and α and β label different energy eigenstates, has been evaluated in the literature in two different ways: by treating the energy eigenfunctions as Gaussian random variables and averaging \|Aαβ\|2 over them; and by relating \|Aαβ\|2 to the classical time-correlation function of A. We show that these two methods give the same answer only if Berry's formula for the spatial correlations in the energy eigenfunctions (which is based on a microcanonical density in phase space) is modified at large separations in a manner that we previously proposed.
- Publication:
-
Physical Review E
- Pub Date:
- June 1998
- DOI:
- 10.1103/PhysRevE.57.7313
- arXiv:
- arXiv:chao-dyn/9711020
- Bibcode:
- 1998PhRvE..57.7313H
- Keywords:
-
- 05.45.+b;
- 03.65.Sq;
- Semiclassical theories and applications;
- Nonlinear Sciences - Chaotic Dynamics;
- Condensed Matter - Mesoscale and Nanoscale Physics
- E-Print:
- 7 pages, no figures, RevTeX