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

TL;DR

This paper enhances the diagonalization method for quantum systems by applying high-order short-time propagator expansions, enabling more accurate eigenvalue and eigenstate calculations in one and two dimensions.

Contribution

It introduces a systematic improvement using effective action for high-order short-time expansions, significantly increasing accuracy in quantum property calculations.

Findings

Achieved high-precision eigenvalues and eigenstates for quantum models.

Validated results through comparison with semiclassical density of states.

Demonstrated effectiveness in one and two-dimensional systems.

Abstract

In this paper, building on a previous analysis [1] of exact diagonalization of the space-discretized evolution operator for the study of properties of non-relativistic quantum systems, we present a substantial improvement to this method. We apply recently introduced effective action approach for obtaining short-time expansion of the propagator up to very high orders to calculate matrix elements of space-discretized evolution operator. This improves by many orders of magnitude previously used approximations for discretized matrix elements and allows us to numerically obtain large numbers of accurate energy eigenvalues and eigenstates using numerical diagonalization. We illustrate this approach on several one and two-dimensional models. The quality of numerically calculated higher order eigenstates is assessed by comparison with semiclassical cumulative density of states.