From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Operator Theory question regarding uncertainty principle
Date: 19 Jan 1999 17:55:59 -0500
Newsgroups: sci.math
Keywords: Heisenberg's inequality in Operator Theory
In article <782sf9$lir$1@nnrp1.dejanews.com>,
wrote:
>Hi,
>
>I have a note here stating that when two operators satisfy the following
>property:
>
>PQ-QP=I
>
>and ||f(t)|| =1 in L^2, that these two facts alone establish
>
>1/2 < ||Pf(t)||*||Qf(t)||
>
>I can prove this in the specific case where Pf = t*f(t) and Qf=df/dt, but
>I can't prove it just from the PQ-QP=I statement.
>
>Can you? Are there other qualifications on the operators necessary? If so,
>what are they?
>
From Dunford and Schwartz: Linear Operators II, Exercise XII.9.40 (edited
for ASCII):
Let A and B be self-adjoint operators in Hilbert space such that
D_0 = dom(AB) intersect dom(BA)
is dense. Let x be in D_0. Write
E(C) = (Cx, x) and V(C) = norm((C-E(C)*I)x)^2
for every C for which Cx is defined.
Then Heisenberg inequality says
V(A) * V(B) >=(1/4) * (abs(E(AB-BA)))^2
So, the note you mentioned has to be amended. I've gone through the
exercise, and it is an application of Cauchy-Schwarz inequality. The
self-adjointness of A and B is important.
The subtraction of E(C)*I in the definition of V(C) is not essential since
AB-BA = [A,B] = [A-p*I, B-q*I] for scalars p, q. It does make good sense,
though, because it minimizes the norm.
Good luck, ZVK(Slavek).