Now we wish to introduce a partial wave decomposition in order to solve the Lippman-Schwinger equation. With a partial wave decomposition we can reduce a three-dimensional integral equation to a one-dimensional one.

Let us continue with our Schroedinger equation in the abstract vector representation $$ \left(T + V\right)\vert\psi_n\rangle = E_n\vert\psi_n \rangle $$ Here \( T \) is the kinetic energy operator and \( V \) is the potential operator. The eigenstates form a complete orthonormal set according to $$ \mathbf{1}=\sum_n\vert\psi_n\rangle\langle\psi_n\vert, \:\: \langle\psi_n\vert\psi_{n'}\rangle =\delta_{n,n'} $$