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\textbf{Homework 2.}
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\noindent
Consider the atomic limit of the Hubbard Model
$$
H=-\mu \sum_{\sigma }f_{\sigma }^{\dagger }f_{\sigma }+Uf_{\uparrow
}^{\dagger }f_{\uparrow }f_{\downarrow }^{\dagger }f_{\downarrow }
$$

\noindent
(1) Compute the retarded Green's function and the spectral function.
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\noindent
(2) Consider the Hubbard operators
\begin{eqnarray*}
X_{\uparrow 0} &=&f_{\uparrow }^{\dagger }(1-n_{\downarrow }) \\
X_{d\downarrow } &=&f_{\uparrow }^{\dagger }n_{\downarrow }
\end{eqnarray*}
referring to atomic states $\left| 0\right\rangle ,\left| \uparrow
\right\rangle ,\left| \downarrow \right\rangle $ and $\left| d\right\rangle
=\left| \uparrow \downarrow \right\rangle \equiv f_{\uparrow }^{\dagger
}f_{\downarrow }^{\dagger }\left| 0\right\rangle $.

\noindent
Express the operator $f_{\uparrow }^{\dagger }$ in terms of Hubbard
operators.

\noindent
(3) Repeat (2) for $f_{\downarrow }^{\dagger }$.

\noindent
(4) Evaluate $\left\langle \left\{ X_{\uparrow 0}X_{0\uparrow }\right\}
\right\rangle $ and $\left\langle \left\{ X_{d\downarrow }X_{\downarrow
d}\right\} \right\rangle $. What is the physical meaning of these quantities
?

\noindent
(5) Evaluate $\left\langle \left\{ X_{\uparrow 0}\left[ HX_{0\uparrow
}\right] \right\} \right\rangle $ and $\left\langle \left\{ X_{d\downarrow
}\left[ HX_{\downarrow d}\right] \right\} \right\rangle $. What is the
physical meaning of these averages ?

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