Computationally efficient method to construct scar functions

Abstract

The performance of a simple method [E. L. Sibert III, E. Vergini, R. M. Benito, and F. Borondo, New J. Phys. 10, 053016 (2008)] to efficiently compute scar functions along unstable periodic orbits with complicated trajectories in configuration space is discussed, using a classically chaotic two-dimensional quartic oscillator as an illustration.

Short-periodic-orbit method for excited chaotic eigenfunctions

“…Different methods have been reported in the literature for the construction of localized states over unstable POs (hereafter referred to as "scar functions" due to their similarity with Heller's scarring on eigenfunctions). Some use averages over groups of eigenfunctions around the PO quantization condition [25]; others make use of the short PO theory [26,27], apply the asymptotic boundary layer [28], or perform the quantum propagation of wave packets launched along the PO of interest [6,29].…”

Using basis sets of scar functions

“…Vergini and coworkers used the short PO theory [22] and obtained scar functions by combination of resonances of POs over which the condition of minimum energy dispersion is imposed, thus including the semiclassical dynamics around the scarring PO up to the Ehrenfest time [23]. Sibert et al [24] and Revuelta et al [25] extended the method to smooth potential systems. Also, Vagog et al extend to unstable POs the asymptotic boundary layer method to calculate stable microresonator localized modes [26].…”

“…In this paper we introduce a new method to construct basis sets formed by the scar functions described before [23][24][25] that can be used to efficiently compute the eigenvalues and eigenfunctions of classically chaotic systems with smooth potentials.…”

Semiclassical basis sets for the computation of molecular vibrational states

“…As already stated in Subsec. III A, the tube/scar functions have a very low dispersion in energy [19,30,35,36,38]. One can then ask whether there is a similar relationship for the eigenfunctions computed in a basis set formed by these localized wave functions.…”

“…For example, Polavieja et al averaged groups of eigenfunctions by performing a short-time quantum evolution [30], and Vergini and coworkers [35] combined PO resonances by minimizing energy dispersion, including then the semiclassical dynamics around the scarring PO up to the Ehrenfest time [36]. More recently, Sibert et al [37] and Revuelta et al [19,38] applied these ideas to systems with smooth potentials, and Vagov et al [39] extended the asymptotic boundary layer method to calculate stable microresonator localized modes over unstable POs.…”