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 Further, for the second case, I get another error that the conditional "if" statement is obsolete.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 ________________________________________________________________________ Want to chat instantly with your online friends? Get the FREE Yahoo! Messenger http://mail.messenger.yahoo.co.uk