Properties of Quantum Systems via Diagonalization of Transition Amplitudes I: Discretization Effects

Abstract

We analyze the method for calculation of properties of non-relativistic 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 followup paper [1] 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.

8 pages, 7 figures, uses revtex4

References in corpus (1)

• Properties of Quantum Systems via Diagonalization of Transition Amplitudes II: Systematic Improvements of Short-time Propagation

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• Fast Converging Path Integrals for Time-Dependent Potentials II: Generalization to Many-body Systems and Real-Time Formalism
• SPEEDUP Code for Calculation of Transition Amplitudes via the Effective Action Approach