The Lippmann-Schwinger equation is often solved by the addition of the factor $i\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle}\newcommand{\e}[0]{\varepsilon} \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\bra}[1]{\left< #1\right|}\e$ such that we have: $$\ket{\psi}=\ket{p}+\f{V}{E-H_0\pm i\e}\ket{\psi}$$ which produces a Green's function of the form (in the 1d case): $$G_\pm=\bra{x}\f{1}{E-H_o\pm i\e}\ket{x'}$$ This is a pain to solve, requiring contour integration. It is much simpler to just solve the equation: $$(H_0-E)G=\delta(x-x')$$ which defines $G$ with the appropriate boundary conditions on $G$. This latter method does not involve the introduction of any factors of $\e$, but I have never seen it done like this. Why not? is this method in someway wrong?