Approximation of Schr{\"o}dinger Unitary Groups of Operators by Particular Projection Methods

TL;DR

This paper develops particular projection methods to approximate Schrödinger unitary groups of operators in quantum systems, providing theoretical insights and numerical estimates for their implementation.

Contribution

It introduces a novel class of projection techniques compatible with quantum theory to approximate operator groups in discretizable Hilbert spaces.

Findings

Established relations between projection methods and original operators.

Provided numerical estimates validating the approximation techniques.

Abstract

In this paper we work with the approximation of unitary groups of operators of the form $e^{-itH}$ where $H\in\mathscr{L}(\mathcal{H})$ is the Hamiltonian of a given quantum dynamical system modeled in the discretizable Hilbert space $\mathcal{H}=\mathcal{H}(G)$, to perform such approximations we implement some techniques from operator theory that we name particular projection methods by compatibility with quantum theory conventions. Once particular representations are defined we study the interelation between some of them properties with the original operators that they mimic. In the end some estimates for numerical implementation are presented to verify the theoretical discussion.