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MathGroup Archive 2003

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RSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg43804] RSolve
  • From: Peter Szabo <peterszabo20022003 at yahoo.co.uk>
  • Date: Mon, 6 Oct 2003 02:07:56 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear Colleagues,

I have a serious problem concerning RSolve for which I
request help.  I have a set of difference equations:


y[n+1]+(a_1 -b*n*n -2)*y[n] + y[n-1]==0. 
y[m+1]+(a_2 -b*m*m -2)*y[m] + y[m-1]==0. 

These are ordinary difference representations (lattice
equations) for the 2-D time independent Schroedinger
equation with harmonic potential.  The condition is
a_1 + a_2=a, which is the coefficient of the partial
difference equation (combined case). Also, "n" and "m"
are the iteration indices (independent variables or
lattice variables) for the 2 dimensions respectively. 

I wish to use "RSolve" to get all possible solutions
for different a_1 and a_2 and b. This includes the
VERY IMPORTANT Gaussian case (discrete Gaussian wave
function) which for 1-D is obtained when
a_1=b/Sqrt[2]. 

I also want to numerically and graphically show that
this is the special case. I have a serious problem
with RSolve that I would request your kind
advise/help. 

When I load RSolve (using Mathematica 4.1) typing
<<DiscreteMath`RSolve`, and then use RSolve[ ] to
solve the discrete system of equations for different
values of a_1, a_2 and b, I GET NO ANSWER. The "Out"
is the same as the "In". Sometimes, I get the error
that this is not a (discrete) equation. This happens
even when I type the case fo 1-D: 

<<DiscreteMath`RSolve` 

RSolve[y[n+1]==-(a_1 - b*n*n -2)*y[n] -y[n-1], 
y[n],n]. 

Specifically, MATHEMATICA either gives NO answer, or
tells me that this is not an equation!!!!. Further,
since the solutions have y[0]=0, even when I type
specifying n>=0 (Greater than or equal to zero): 

RSolve[{y[n+1]==-(a_1 - b*n*n -2)*y[n] -y[n-1]/;n>=1,
y[0]==0},y[n],n], 

I get the same result--either NO answer or an eror
that this is NOT AN EQUATION. Further, for the second
case, I get another error that the conditional "if"
statement is obsolete. Can you please help me here.
Anything I try in solving these discrete equations
with RSolve and then plot the solutions do NOT work. 

I WISH TO ADD THAT THE PROBLEM DOES NOT GO AWAY EVEN
BY SETTING n->n+1.

Could some kind colleague please help me out.

Most Respectfully Yours

Peter Szabo

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