A Problem with Lippmann-Schwinger Equation

[Hyperreality]

I am doing Problem 3.1 from Quantum Theory of Fields by Steven Weinberg regaring the Lippmann-Schwinger Equation. The problem states,

Given a separable interaction,

$$\langle \alpha_{0}|V|\beta_{0}\rangle=g u_{\alpha}u_{\beta}^{*}$$

where $$g$$ is a real coupling constant, and $$u_{\alpha}$$ is a set of complex quantities with

$$\sum_{\alpha}|u_{\alpha}|^{2}=1$$

Use the Lippmann-Schwinger equation to find explicit solutions for the 'in' and 'out' state and the S-matrix.

I believe I have solved this to the second order in $$g$$, but I'm not sure if it is correct since there are no solutions. What I found puzzling is that my S-matrix has 0 contribution for the first order $$g$$. Is this suppose to happen? I do not want to show the working since it is too much mathematics. I am just wondering if there are anyone who has done similar problems.

Any comment is appreciated!

 

[SayanMandal]

I believe I have solved this to the second order in $$g$$, but I'm not sure if it is correct since there are no solutions. What I found puzzling is that my S-matrix has 0 contribution for the first order $$g$$. Is this suppose to happen? I do not want to show the working since it is too much mathematics. I am just wondering if there are anyone who has done similar problems.

Any comment is appreciated!

I am facing a similar problem too. I solved it up to 2nd order, and there was no contribution to the S-matrix at $$O(g)$$ Also, I was wondering if there are non-approximate exact solutions. Did you find anything else?