In order to get ideas for unique research topics, as well as to become more knowledgeable about current research, I have been reading published papers and have been working to reproduce their results.  One such paper was written by Gerald Dunne and Jake Mannix entitled "Supersymmetry Breaking with Periodic Potentials," published in Physics Letters B428, 1998, pages 115-118.  The results below are taken from this paper.

W(x) = tanh (x) - a

Y minus for a = 0.5

Y plus

The zero modes of the minus Hamiltonian are normalizable for -1<a<1. The zero modes of the plus Hamiltonian are not normalizable.

W(x) = (tanh (x))^2 - a

Y minus

Y plus

The above superpotential generalized in Jacobi elliptic functions is given by W(x) = m2*sn2(x|m)*cd2(x|m) - a.  The paper makes the claim that the integral of W(x) is given by:

(2-m-a)x+m*sn(x|m)*cd(x|m)-2*E(x|m).

However, the two graphs below contract.  The first is a graph of a numerical integration of W(x).  The second is a plot of the above function.  The values of the parameters, m and a, are the same in both cases.

In order to find the error in the paper, the integral of the superpotential had to be done by hand (considering that Mathematica wouldn't do it).  It was discovered that the error was in sloppy, careless notation on the part of the author.  The actual integral of the superpotential is:

(2-m-a)x+m*sn(x|m)*cd(x|m)-2*E(am(x|m)|m), or

(2-m-a)x+m*sn(x|m)*cd(x|m)-2*E(|m).

Current work has been an extension of the ideas presented in this paper.

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