Commutators and Operator Powers in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg16635] Commutators and Operator Powers in Mathematica
- From: Alan Lewis <alanlewis at home.com>
- Date: Fri, 19 Mar 1999 12:53:55 -0500
- Organization: @Home Network
- Sender: owner-wri-mathgroup at wolfram.com
I am looking for any links or suggestions on implementing
commutation relations and powers of differential operators
in mathematica.
As an example, I have two operators L0 and L1 that act on arbitrary
(well say infinitely differentiable) functions f[x]
L0 simply multiplies f[x] by x.
L1 = a x^(3/2) D[f[x],x] + b x^2 D[f[x],{x,2}]
where a,b are constants independent of x. The second line is not
meant to be working math. code but is just meant to explain the action
of this differential operator.
Now what I want to do is be able to evaluate repeated commutators
and powers of these operators. For example, the first commutator
should evaluate to:
[L0,L1]f[x] = x L1 f[x] - L1 (x f[x]) =
-a x^(3/2) f[x] - 2 b x^2 D[f[x],x]
I would also like to evaluate powers such as
L1^n, meaning the operator acts on f[x] n times. Repeated
commutators are expressions like
[L1,[L0,L1]] or [L0,[L0,L1]], etc.
The action of L1 is just an example, but the general class of operators
I am interested in are always the sum of a first and second derivative
with simple expressions like the above in front of the derivative.
And L0 is always multiplication by x.
Thanks in advance for any suggestions,
Alan
- Follow-Ups:
- Re: Commutators and Operator Powers in Mathematica
- From: "Nguyen N. Anh" <anh@chm.ulaval.ca>
- Re: Commutators and Operator Powers in Mathematica