7. In the Cartesian coordinate system $x O y$, the ellipse $\Gamma: \frac{x^{2}}{3}+y^{2}=1$ has its left vertex at $P$, and $A, B$ are two moving points on $\Gamma$. Then the range of $\overrightarrow{P A} \cdot \overrightarrow{P B}$ is $\qquad$.
Let $A(\sqrt{3} \cos \alpha, \sin \alpha), B(\sqrt{3} \cos \beta, \sin \beta), P(-\sqrt{3}, 0)$,
then $\overrightarrow{P A} \cdot \overrightarrow{P B}=(\sqrt{3} \cos \alpha+\sqrt{3})(\sqrt{3} \cos \beta+\sqrt{3})+\sin \alpha \sin \beta$
$$
\begin{array}{l}
=3 \cos \alpha \cos \beta+3(\cos \alpha+\cos \beta)+\sin \alpha \sin \beta+3 \\
=\cos (\alpha+\beta)+2 \cos (\alpha-\beta)+6 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}+3 \\
=2 \cos ^{2} \frac{\alpha+\beta}{2}+4 \cos ^{2} \frac{\alpha-\beta}{2}+6 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2},
\end{array}
$$

Let $u=\cos \frac{\alpha+\beta}{2}, v=\cos \frac{\alpha-\beta}{2}$, then $u, v \in[-1,1]$,
$\overrightarrow{P A} \cdot \overrightarrow{P B}=2 u^{2}+6 u v+4 v^{2}=4\left(v+\frac{3}{4} u\right)^{2}-\frac{1}{4} u^{2} \in\left[-\frac{1}{4}, 12\right]$.
When $\overrightarrow{P A} \cdot \overrightarrow{P B}=-\frac{1}{4}$, $A\left(-\frac{3 \sqrt{3}}{4}, \frac{\sqrt{7}}{4}\right), B\left(-\frac{3 \sqrt{3}}{4},-\frac{\sqrt{7}}{4}\right)$;
When $\overrightarrow{P A} \cdot \overrightarrow{P B}=12$, $A(\sqrt{3}, 0), B(\sqrt{3}, 0)$. Therefore, the range of $\overrightarrow{P A} \cdot \overrightarrow{P B}$ is
$$
\left[-\frac{1}{4}, 12\right] \text {. }
$$