The Howland - Kato Commutator Problem

TL;DR

This paper explores the conditions under which the commutator of functions of position and momentum operators is positive, addressing a longstanding problem in operator theory with partial results and discussions.

Contribution

It provides new partial answers and insights into Kato's conjecture regarding the positivity of certain operator commutators.

Findings

Partial conditions for positivity of commutators

Discussion on Kato's conjecture and its validity

New theoretical insights into operator commutator problems

Abstract

We investigate the following problem: For what $f$ and $g$ is the commutator $i[f(P),g(Q)]$ positive when $f$ and $g$ are bounded measurable functions? This problem originated in work of James Howland and was pursued by Tosio Kato who suggested what might be the answer. So far there is no proof that Kato was correct but in this paper we discuss the problem and give some partial answers to the above question.