Relevance of chaos in numerical solutions of quantum billiards

Abstract

In this paper we have tested several general numerical methods in solving the quantum billiards, such as the boundary integral method (BIM) and the plane-wave decomposition method (PWDM). We performed extensive numerical investigations of these two methods in a variety of quantum billiards: integrable systems (circles, rectangles, and segments of a circular annulus), Kolmogorov-Arnold-Moser systems (Robnik billiards), and fully chaotic systems (ergodic, such as a Bunimovich stadium, Sinai billiard, and cardiod billiard). We have analyzed the scaling of the average absolute value of the systematic error $\ensuremath{\Delta}E$ of the eigenenergy in units of the mean level spacing with the density of discretization $b$ (which is the number of numerical nodes on the boundary within one de Broglie wavelength) and its relationship with the geometry and the classical dynamics. In contradistinction to the BIM, we find that in the PWDM the classical chaos is definitely relevant for the numerical accuracy at a fixed density of discretization $b$. We present evidence that it is not only the ergodicity that matters, but also the Lyapunov exponents and Kolmogorov entropy. We believe that this phenomenon is one manifestation of quantum chaos.