RELEVANCE OF CHAOS IN NUMERICAL-SOLUTIONS OF QUANTUM BILLIARDS

In this paper we have tested several general numerical methods in solv ing 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, Si nai billiard, and cardiod billiard). We have analyzed the scaling of t he average absolute value of the systematic error Delta E of the eigen energy in units of the mean level spacing with the density of discreti zation b (which is the number of numerical nodes on the boundary withi n one de Broglie wavelength) and its relationship with the geometry an d the classical dynamics. In contradistinction to the BIM, we find tha t in the PWDM the classical chaos is definitely relevant for the numer ical accuracy at a fixed density of discretization b. We present evide nce that it is not only the ergodicity that matters, but also the Lyap unov exponents and Kolmogorov entropy. We believe that this phenomenon is one manifestation of quantum chaos.