A question on the transition matrix. (algebraic derivation)

Question

I am trying to follow the derivation of the following identity:

$$[\epsilon -H_0+i\eta]^{-1}T = [\epsilon - H +i\eta]^{-1}U$$

where $T$ is the transition matrix and $U$ is the potential caused by the impurities.

I have also the next identity: $$T=U+U[\epsilon-H_0+i\eta]^{-1}T$$ which I thought to use in order to prove the first identity. In the textbook called A Quantum Approach to Condensed Matter Physics it's said that the identity: $$[\epsilon -H_0+i\eta]^{-1}T = [\epsilon - H +i\eta]^{-1}U$$ can be seen by operating on both sides with $\epsilon-H+i\eta$ which I did but I don't see how to get to an identity which is known already before this identity. A few more details this derivation is on pages 172-173 of the above book and also $\eta \to 0$.

I appreciate your help.

P.S $$H=H_0+U$$ I think though it's not written explicitly in the book.

Answer 1

First, express $T$ from your second equation: $$ T = \left[I - U[\epsilon -H_0 +i\eta]^{-1}\right]^{-1} U, $$ where $I$ is the identity operator. Second, using properties of an inverse operator, write identities: $$ \left[I - U[\epsilon -H_0 +i\eta]^{-1}\right]^{-1} = \left[[\epsilon -H_0+i\eta -U][\epsilon-H_0+i\eta]^{-1}\right]^{-1} = [\epsilon-H_0+i\eta][\epsilon -H_0+i\eta -U]^{-1} $$ Now we have obtained $$ T = [\epsilon-H_0+i\eta][\epsilon -H_0+i\eta -U]^{-1}U $$ As $H_0+U = H$, it is straightforward to obtain the needed equation from the last one.