Operator integrals and sesquilinear forms

TL;DR

This paper explores methods for defining unbounded operator integrals with respect to positive measures and sesquilinear forms, analyzing their relationships and implications in quantum operator theory.

Contribution

It introduces systematic approaches to operator integrals and examines their connections within the extension theory of symmetric operators, supported by a physically relevant example.

Findings

Different definitions of operator integrals are shown to be related.

Mathematical subtleties are highlighted through a quantum example.

The work advances understanding of unbounded operator measures in quantum physics.

Abstract

We consider various systematic ways of defining unbounded operator valued integrals of complex functions with respect to (mostly) positive operator measures and positive sesquilinear form measures, and investigate their relationships to each other in view of the extension theory of symmetric operators. We demonstrate the associated mathematical subtleties with a physically relevant example involving moment operators of the momentum observable of a particle confined to move on a bounded interval.