The equation ⟨ϕm|ˆT|ϕn⟩=⟨ϕm|ˆV|ϕn⟩+⟨ϕm|ˆV1(En−ˆH0)ˆT|ϕn⟩, is called the Lippman-Schwinger equation. Inserting the completeness relation 1=∑n|ϕn⟩⟨ϕn|,⟨ϕn|ϕn′⟩=δn,n′ we have ⟨ϕm|ˆT|ϕn⟩=⟨ϕm|ˆV|ϕn⟩+∑k⟨ϕm|ˆV|ϕk⟩1(En−ωk)⟨ϕk|ˆT|ϕn⟩, which is (when we specify the state |ϕn⟩) an integral equation that can actually be solved by matrix inversion easily! The unknown quantity is the T-matrix.