A duality theory for unbounded Hermitian operators in Hilbert space

TL;DR

This paper introduces a duality framework for unbounded Hermitian operators in Hilbert spaces, providing new insights into their deficiency indices and selfadjoint extensions, which are crucial in quantum mechanics and functional analysis.

Contribution

It develops a duality theory that simplifies the analysis of deficiency spaces for Hermitian operators by relating them to dual operators, aiding in understanding their selfadjoint extensions.

Findings

Provides a new duality approach to compute deficiency spaces

Simplifies the analysis of selfadjoint extensions

Offers practical tools for operators in quantum mechanics

Abstract

We develop a duality theory for unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify additional structures that throw light on the problem. While duality considerations are a tested tool in mathematics, we will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space.