Re: Re: Commutators and Operator Powers in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg16744] Re: [mg16692] Re: Commutators and Operator Powers in Mathematica
- From: Daniel Lichtblau <danl>
- Date: Wed, 24 Mar 1999 02:23:54 -0500
- References: <19990320225455.005056@post.demon.co.uk>
- Sender: owner-wri-mathgroup at wolfram.com
Andrzej Kozlowski wrote: > > I was about to sent my answer when I noticed that Daniel Lichtblau had > beaten me to it and I could not hope to improve on his solution. So I > stopped writing mine but then I still found one thing about his answer > that was not entirely satisfactory from my point of view. I like to work > with pure functions rather than expressions involving some arbitrarily > chosen variables like x etc. I find this approach both aesthetically more > satisfactory and also better corresponding to the way I think of > structures such as function and operator algerbas etc. So I re-wrote > Daniel's package to eliminate the need to refer to any "vars" and decided > to post it in case there are other people who share my bias in this matter. > > I first attatch new rules to Plus, Times and Power to allow algebraic > operations on pure functions: > > Unprotect[{Plus, Times,Power}]; > Plus/: ((f_) + (g_))[x_] := f[x] + g[x]; > Times/: ((k_?NumberQ )(f_))[x_] := k f[x]; > Times/: ((f_) (g_))[x_] := f[x] g[x]; > Power/:(f_^n_)[x_]:=f[x]^n ; > Protect[{Plus,Times,Power}]; > > Next we re-write Daniel's definitions eliminating all vars: > > In[2]:= > differentialOperate[a_, fn_] /; FreeQ[a, D] := a*fn > differentialOperate[L1_ + L2_, fn_] := > differentialOperate[L1, fn] + differentialOperate[L2,fn] > differentialOperate[a_*L_,fn_] /; FreeQ[a, D] := > a*differentialOperate[L,fn] > differentialOperate[D^(n_.),fn_] := Derivative[n][fn] > differentialOperate[L1_**L2_,fn_] := > differentialOperate[L1, differentialOperate[L2,fn]] > differentialOperate[L1_**L2_**L3__, fn_] := > differentialOperate[L1, differentialOperate[L2**L3,fn]] > differentialOperate[bracket[L1_,L2_],fn_] := Expand[ > differentialOperate[L1, differentialOperate[L2,fn]] - > differentialOperate[L2, differentialOperate[L1,fn]]] > differentialOperate[L1_^(n_.), fn_] := > Nest[Expand[differentialOperate[L1,#]]&,fn, n] > > Now we define our operators using pure functions as coefficients: > > In[3]:= > L0 = #&; > L1 = (a*#^(3/2)&)*D +( b*#^2&)*D^2; > > Now, given a function f, we can define the > > In[11]:= > bracket[L0,L1][f]:= > Function[t,differentialOperate[bracket[L0,L1],f][t]//Simplify] > > This indeed is the right function, e.g. > > In[12]:= > bracket[L0,L1][f][t] > Out[12]= > 3/2 2 > -a t f[t] - 2 b t f'[t] > This variable-free approach can be useful but let me point out a few possible pitfalls. First, by attaching rules to Plus et al there is always the danger of unintended side effects during evaluation. Next, you now do not have a way to handle multivariate functions very well, because a "variable" will be implied. For example, you get In[50]:= bracket[L0,L1][f][t,x] 3/2 2 Out[50]= -(a t f[t]) - 2 b t f'[t] In[51]:= bracket[L0,L1][f][x,t] 3/2 2 Out[51]= -(a x f[x]) - 2 b x f'[x] whereas using the admittedly imperfect code I sent one has: In[13]:= differentialOperate[bracket[L0,L1], f[t,x], x] 3/2 2 (0,1) Out[13]= -(a x f[t, x]) - 2 b x f [t, x] and something similar if we instead specify 't' to be the variable of interest. Moreover nested functions present no problem: In[14]:= differentialOperate[bracket[L0,L1], f[g[x],x,t], x] 3/2 2 (0,1,0) Out[14]= -(a x f[g[x], x, t]) - 2 b x f [g[x], x, t] - 2 (1,0,0) > 2 b x g'[x] f [g[x], x, t] One aspect to my code I do not like is the fact that takind mixed derivatives with respect to different variables is at best awkward. So amybe it would be better to carry this in the D[...] parts of the differential operators as a pure function? This is something I need to ponder. Daniel Lichtblau Wolfram Research > On Sat, Mar 20, 1999, Daniel Lichtblau <danl at wolfram.com> wrote: > > >Alan Lewis wrote: > >> > >> 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 > > > > > >Here is some code to tangle with differential operators. > > > >differentialOperate[a_, expr_, var_] /; FreeQ[a, D] := a*expr > >differentialOperate[L1_ + L2_, expr_, var_] := > > differentialOperate[L1, expr, var] + differentialOperate[L2, expr, > >var] > >differentialOperate[a_*L_, expr_, var_] /; FreeQ[a, D] := > > a*differentialOperate[L, expr, var] > >differentialOperate[D^(n_.), expr_, var_] := D[expr, {var, n}] > >differentialOperate[L1_**L2_, expr_, var_] := > > differentialOperate[L1, differentialOperate[L2, expr, var], var] > >differentialOperate[L1_**L2_**L3__, expr_, var_] := > > differentialOperate[L1, differentialOperate[L2**L3, expr, var], var] > >differentialOperate[bracket[L1_,L2_], expr_, var_] := Expand[ > > differentialOperate[L1, differentialOperate[L2, expr, var], var] - > > differentialOperate[L2, differentialOperate[L1, expr, var], var]] > >differentialOperate[L1_^(n_.), expr_, var_] := > > Nest[Expand[differentialOperate[L1,#,var]]&, expr, n] > > > > > >For your examples, one has > > > >L0 = x; > >L1 = a*x^(3/2)*D + b*x^2*D^2; > > > >Then > > > >In[12]:= differentialOperate[bracket[L0,L1], f[x], x] > > > > 3/2 2 > >Out[12]= -(a x f[x]) - 2 b x f'[x] > > > > > >In[21]:= e1 = Expand[differentialOperate[L1, f[x], x]] > > > > 3/2 2 > >Out[21]= a x f'[x] + b x f''[x] > > > >In[22]:= e2 = Expand[differentialOperate[L1, e1, x]] > > > > 3/2 2 2 > > 3 a b x f'[x] 3 a x f'[x] 2 2 > >Out[22]= ---------------- + ------------- + 2 b x f''[x] + > > 4 2 > > > > 5/2 2 3 2 3 (3) > >> 5 a b x f''[x] + a x f''[x] + 4 b x f [x] + > > > > 7/2 (3) 2 4 (4) > >> 2 a b x f [x] + b x f [x] > > > >In[23]:= differentialOperate[L1^2, f[x], x] == e2 > > > >Out[23]= True > > > > > >A slightly cruder version of this may all be found in a notebook I > >prepared for the 1998 Worldwide Mathematica Users Conference, a copy of > >which can be found at: > > > >http://www.wolfram.com/conference98/schedule/symbolic_FAQ.html > > > >in the section "Some noncommutative algebraic manipulation". There is > >also some code therein to do algebraic simplification in a commutator > >algebra which may be of relevance. > > > > > >Daniel Lichtblau > >Wolfram Research > > Andrzej Kozlowski > Toyama International University > JAPAN > http://sigma.tuins.ac.jp/ > http://eri2.tuins.ac.jp/