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

TL;DR

This paper investigates a numerical method for quantum systems that uses diagonalization of short-time evolution operators, demonstrating that it significantly reduces discretization errors compared to traditional Hamiltonian diagonalization, especially for few-body systems.

Contribution

The paper introduces a discretization error analysis showing exponential decay of errors in a diagonalization-based approach, improving numerical accuracy for quantum system simulations.

Findings

Discretization errors decrease exponentially with 1/Δ^2 in the proposed method.

The approach is particularly effective for few-body quantum systems.

Numerical results confirm the analytical error estimates across multiple models.

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.