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lattice Schroedinger equation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg43807] lattice Schroedinger equation
*From*: peterszabo20022003 at yahoo.co.uk (Peter Szabo)
*Date*: Mon, 6 Oct 2003 02:07:58 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Dear Colleagues;
I request help in solving the following system of discrete/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 discrete/ordinary difference representations (lattice
equations) for the 2-D time independent discrete Schroedinger-like
equation with harmonic potential obtained by a separation of variables
ansatz.
Here, "a_1", "a_2", and "b" are the eigen constants. 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 have the
following requirements:
1. I would like to plot the system fo all possible admissable values
of y[n],..., and constants "a_k" abd"b".
2. VITAL: I would also like to simulate and plot the IMPORTANT case
for Gaussian wave functions, that occur (for example in the 1-D case
when a_1=b/Sqrt[2]. Further, I would like to verify the relationship
between constants a_1,...,a_p and "b" for a p-dimensional case.
Another equation I would like to solve is a simple form of the first
discrete Painleve equation:
y[n+1]+y[n]+y[n-1]+((a*n +b)/(1+y[n])) +mu==0.
Here, "a", "b" and "mu" are constants. As you very
well know, this is a simple modification of the
example given by Eq. (3.3.1) in B. Grammaticos, F. W.
Nijhoff and A. Ramani, "Discrete Pailleve Equations",
Lecture Notes for the Cargese School, (1996).
The only modifications were that the third constant
"gamma" is set to zero and the translation
y[n]->y[n+1] is done in the denominator of the fourth
term above, that is associated with the constants.
This is done fo the case of simplification.
Could anyone PLEASE help me out in this?
Most Respectfully Yours
Peter Szabo
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