By $L(V_{\alpha})$, I mean the least inner model containing $V_{\alpha}$.
I'm especially interested in e.e's $j:L(V_{\alpha}) \to L(V_{\beta})$, where $ \omega < \vert \beta \vert < \vert \alpha \vert $.
I know Laver did some work where $ \alpha = \beta = \lambda + 2 $ for a singular cardinal $\lambda$ with countable cofinality.