Vol. 95, No. 2, 1981

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ISSN: 0030-8730
On the relation PQ QP = iI

W. J. Phillips

Vol. 95 (1981), No. 2, 435–441
Abstract

There is a variety of literature on the relationship between the two equations

PQ − QP ⊂  − iI                     (1.1)

exp(itP )exp(isQ ) = exp(ist)exp(isQ )exp(itP), s,t ∈ R     (1.2)

where P and Q are self-adjoint operators on a Hilbert space. Von Neumann has characterized the solutions of (1.2) as those pairs (P,Q) which are unitarily equivalent to a direct sum of a number of copies of the Schrödinger pair (p,q) where p is i(d∕dx) and q is multiplication by x on L2(R). Hence any pair which satisfies (1.2) and is irreducible in the obvious sense is unitarily equivalent to the Schrödinger pair. It is well known that any pair satisfying (1.2) satisfies (1.1) in the following strong sense:

  1. there is a dense subspace Ω which is a core for both P and Q, is invariant under P and Q and PQf QPf = if for all f in Ω.

Mathematical Subject Classification 2000
Primary: 81D05, 81D05
Secondary: 47B25
Milestones
Received: 17 April 1979
Revised: 15 April 1980
Published: 1 August 1981
Authors
W. J. Phillips