A counterexample to the Kato conjecture for positive commutators

Rupert L. Frank, Paata Ivanisvili

arXiv:2608.07805v1 Primary: math.FA Secondary: math-ph, math.OA, math.SP

Abstract

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

Disclosure

“ect-ID 470903074. P.I. acknowledges partial support from the US NSF CAREER grant DMS-2152401, US NSF grant DMS-2554183, a Simons Fellowship, and a Humboldt Research Fellowship for Experienced Researchers. The authors acknowledge the use of AI tools. All mathematical arguments and proofs in the final manuscript were checked and written by the authors. 14”
PDF page 14

Classification: Drafting limited passages
Multiplier: 5
Verified: 2026-08-23T05:00:38.177455Z

Structural counts

Pages 15 pdf

Theorems 1 source

Lemmas 7 source

Propositions 3 source

Corollaries 1 source

Definitions 0 source

Displayed equations 119 source

Bibliography entries 7 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main_v4.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.