Weak commutation relations of unbounded operators and applications

TL;DR

This paper compares four definitions of the commutation relation for unbounded operators, explores their implications for eigenvector existence, and studies the algebraic structures generated, with applications in quantum operator theory.

Contribution

It introduces and compares four weak commutation definitions for unbounded operators, analyzing their consequences and applications in operator algebra and quantum mechanics.

Findings

Different definitions of weak commutation have distinct implications.

Existence of eigenvectors depends on the type of commutation relation.

Partial O*-algebra generated by operators is characterized.

Abstract

Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.