Properties of quantum systems via diagonalization of transition amplitudes. I. Discretization effects

Abstract

We analyze the method for calculation of properties of nonrelativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors associated with space discretization. Approaches using direct diagonalization of real-space discretized Hamiltonians lead to polynomial errors in discretization spacing Delta . Here we show that the method based on the diagonalization of the short-time evolution operators leads to substantially smaller discretization errors, vanishing exponentially with 1/Delta(2). As a result, the presented calculation scheme is particularly well suited for numerical studies of few-body quantum systems. The analytically derived discretization errors estimates are numerically shown to hold for several models. In the follow up paper [I. Vidanović, A. Bogojević, A. Balaz, and A. Belić, Phys. Rev. E 80, 066706 (2009)] we present and analyze substantial improvements that result from the merger of this approach with the recently introduced effective-action scheme for high-precision calculation of short-time propagation.