We will assume that the Hamiltonian may be written as
where  is the free Hamiltonian (or more generally, a Hamiltonian with known eigenvectors).  For example, in nonrelativistic quantum mechanics  may be
Intuitively  is the interaction energy of the system.  Let there be an eigenstate of :
Now if we add the interaction  into the mix, the Schrödinger equation reads
Now consider the Hellmann–Feynman theorem, which requires the energy eigenvalues of the Hamiltonian to change continuously with continuous changes in the Hamiltonian.  Therefore, we wish that  as .  A naive solution to this equation would be
where the notation  denotes the inverse of .  However  is singular since  is an eigenvalue of .  As is described below, this singularity is eliminated in two distinct ways by making the denominator slightly complex:
By insertion of a complete set of free particle states,
the Schrödinger equation is turned into an integral equation. The "in"  and "out"  states are assumed to form bases too, in the distant past and distant future respectively having the appearance of free particle states, but being eigenfunctions of the complete Hamiltonian. Thus endowing them with an index, the equation becomes