Journal article
A Uniqueness Theorem for the Heisenberg-Weyl Commutation Relations with Non-Selfadjoint Position Operator
American journal of mathematics, Vol.103(2), pp.273-287
04/1981
DOI: 10.2307/2374217
Abstract
Let P and Q denote the quantum mechanical momentum and position operators. Let Q0 be a closed operator in L2(R) with dense domain, which is a restriction of Q. For example, for every closed subset $\Lambda \subset \mathbf{R}$ of measure zero, such an operator may be specified by (*)D(Q0) = {f ε D(Q): f̂(λ) = 0 for all λ ε Λ} where D denotes the domain. Restriction operators Q0 arise in the rejpresentation of scattering. We show that every invariant restriction operator Q0 is given by formula (*) where Λ is unique. In the second part we make the connection to spectral analysis in Banach algebras. The Sobolev space H1(R) is naturally associated to a certain unital semisimple Banach algebra B. As a corollary to the uniqueness theorem, we show that spectral synthesis is possible in B. An analogous uniqueness result for $H^m (\mathbf{R}), m > 1$, is included.
Details
- Title: Subtitle
- A Uniqueness Theorem for the Heisenberg-Weyl Commutation Relations with Non-Selfadjoint Position Operator
- Creators
- Palle E. T Jorgensen
- Resource Type
- Journal article
- Publication Details
- American journal of mathematics, Vol.103(2), pp.273-287
- DOI
- 10.2307/2374217
- ISSN
- 0002-9327
- eISSN
- 1080-6377
- Language
- English
- Date published
- 04/1981
- Academic Unit
- Mathematics
- Record Identifier
- 9983985853602771
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