Non-existence of Taylor expansion in time due to cusps

TL;DR

This paper investigates how cusps in electronic density cause non-analytic time behavior, challenging the applicability of Taylor expansions in time-dependent quantum systems, and introduces the s-expansion method for analyzing such cases.

Contribution

It introduces the s-expansion method to analyze non-analytic time behavior caused by cusps in electronic density, providing a new approach to short-time dynamics in quantum systems.

Findings

s-expansion captures dominant short-time behavior

Non-analyticities arise from electronic density cusps

Method applies to various perturbations and dimensions

Abstract

In the usual treatment of electronic structure, all matter has cusps in the electronic density at nuclei. Cusps can produce non-analytic behavior in time, even in response to perturbations that are time-analytic. We analyze these non-analyticities in a simple case from many perspectives. We describe a method, the s-expansion, that can be used in several such cases, and illustrate it with a variety of examples. These include both the sudden appearance of electric fields and disappearance of nuclei, in both one and three dimensions. When successful, the s-expansion yields the dominant short-time behavior, no matter how strong the external electric field, but agrees with linear response theory in the weak limit. We discuss the relevance of these results to time-dependent density functional theory.