Is there relevance of chaos in numerical solutions of quantum billiards?

Abstract

In numerically solving the Helmholtz equation inside a connected plane domain with Dirichlet boundary conditions (the problem of the quantum billiard) one surprisingly faces enormous difficulties if the domain has a problematic geometry such as various nonconvex shapes. We have tested several general numerical methods in solving the quantum billiards. Following our previous paper (Li and Robnik 1995) where we analyzed the Boundary Integral Method (BIM), in the present paper we investigate systematically the so-called Plane Wave Decomposition Method (PWDM) introduced and advocated by Heller (1984, 1991). In contradistinction to BIM we find that in PWDM the classical chaos is definitely relevant for the numerical accuracy at fixed density of discretization on the boundary $b$ ($b$ = number of numerical nodes on the boundary within one de Broglie wavelength). This can be understood qualitatively and is illustrated for three one-parameter families of billiards, namely Robnik billiard, Bunimovich stadium and Sinai billiard. We present evidence that it is not only the ergodicity which matters, but also the Lyapunov exponents and Kolmogorov entropy. Although we have no quantitative theory we believe that this phenomenon is one manifestation of quantum chaos. PACS numbers: 02.70.Rw, 05.45.+b, 03.65.Ge, 03.65.-w