Weak commutation relations of unbounded operators: nonlinear extensions

TL;DR

This paper investigates the implications of weak commutation relations between unbounded operators and explores nonlinear extensions, advancing understanding of their mathematical and physical properties.

Contribution

It introduces a framework for analyzing weak commutation relations and proposes a novel nonlinear extension, expanding the theoretical landscape of operator algebra.

Findings

Analysis of weak commutation relations for unbounded operators

Introduction of nonlinear extensions of these relations

Potential implications for quantum physics and operator theory

Abstract

We continue our analysis of the consequences of the commutation relation $[S,T]=\Id$, where $S$ and $T$ are two closable unbounded operators. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. {We also consider what we call, adopting a physical terminology}, a {\em nonlinear} extension of the above commutation relations.