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 ________________________________________________________________________