# Self-energy that does not obey sum rule

Analytically, I calculated a self-energy $$\Sigma(\omega)$$, for which I verified that

1) $$\text{Im}\big[\Sigma(\omega)\big] \leq 0$$ for all $$\omega$$ and specifically $$\text{Im}\big[\Sigma(0)\big] = 0$$,

2) $$\text{Im}\big[\Sigma(\omega)\big] = 0$$ for $$|\omega|>\omega_C$$,

3) $$\text{Re}\big[\Sigma(\omega)\big] \sim \frac{1}{\omega}$$ for $$\omega\rightarrow\infty$$,

4) $$\text{Re}\big[\Sigma(\omega)\big]$$ and $$\text{Im}\big[\Sigma(\omega)\big]$$ obey the Kramers-Kronig relations.

Besides the analytical calculation, I did a numerical calculation yielding the same results.

I was under the impression that obeying these requirements would be enough to expect the sum rule, i.e., $$\int_{-\infty}^{\infty}d\omega \frac{-2\text{Im}\big[\Sigma(\omega)\big]}{\big(\omega - \text{Re}\big[\Sigma(\omega)\big]\big)^2 + \big(\text{Im}\big[\Sigma(\omega)\big]\big)^2} = 1,$$ to be satisfied. Unfortunately, it is not satisfied in my case.

Hence my question: I thought that an analytic self-energy that vanishes for large $$\omega$$ and has no poles in the uppper half plane always satisfies the sum rule. Apparently the above 4 results are not strong enough to conclude that the sum rule has to be obeyed. What am I missing here?