Using basis sets of scar functions

Abstract

We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two-dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presented.

Computationally efficient method to construct scar functions

“…This is an important point, which indicates that they are suitable candidates to be used as a good dynamical basis set for the efficient computation of eigenstates of this kind of system. We have actually verified this possibility, and the corresponding results will be published in the near future [36].…”

Dynamic localization in walking droplets?

“…In this case, the shape of the motion in configuration space resembles that of those paper clips designed to hold a large number of paper sheets. Actually, this shape is also found as a periodic orbit in the Hamiltonian quartic oscillator [33]. …”

“…In figure 9a-c the localization takes place on the figure of eight motion described in figures 3 and 4, while in figure 9d an accumulation on the so-called square box type periodic orbit (see Ref. [33]) is seen.…”

Semiclassical basis sets for the computation of molecular vibrational states

“…To define our basis set, we have generalized the usual Gram-Schmidt method (GSM) [58], and developed a new selective Gram-Schmidt method (SGSM). This SGSM is the second pillar of our method, and it is able to choose a basis set of linearly independent functions in a vectorial space from a larger (overcomplete) set of functions, that can be used to efficiently compute the chaotic eigenfunctions of our system [40].…”

“…It should be remarked here that for the system under study, similar results would be obtained using solely the tube wave functions. Moveover, they are also adequate for systems with a higher degree of chaoticity [40]. However, we have decided to use the scar wave functions over the unstable POs as they have a lower dispersion in energy, rendering thus slightly better results.…”