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

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.

💡 Research Summary

The paper introduces a novel numerical scheme for computing properties of non‑relativistic quantum systems by directly diagonalizing the short‑time evolution operator (the Euclidean propagator) on a spatial lattice. Traditional approaches discretize the Hamiltonian in real space and then diagonalize the resulting matrix. While straightforward, this method suffers from polynomial discretization errors that scale as a power of the lattice spacing Δ (typically (O(\Delta^{2})) or (O(\Delta^{4}))), limiting achievable precision unless an extremely fine grid is employed.

In contrast, the authors propose to work with the operator (\hat U(\Delta\tau)=\exp(-\Delta\tau\hat H)), where (\Delta\tau) is a short imaginary‑time step. By discretizing space and constructing the matrix representation of (\hat U), they exploit the fact that the propagator already incorporates high‑order short‑time dynamics. A rigorous error analysis, based on a combination of Trotter‑Suzuki factorization, Taylor/Bernoulli expansions, and path‑integral techniques, shows that the dominant discretization error decays exponentially with the inverse square of the lattice spacing:
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