11. Let $F$ be the left focus of the hyperbola $\Gamma: x^{2}-y^{2}=1$, and let a line passing through $F$ intersect $\Gamma$ at points $M$ and $N$.
(1) If $M$ and $N$ are both on the left branch of the hyperbola, find the minimum value of the area of $\triangle O M N$;
(2) Does there exist a point $P$ on the $x$-axis such that $\overrightarrow{P M} \cdot \overrightarrow{P N}$ is a constant? If so, find the coordinates of point $P$; if not, explain why.
(1) As shown in the figure, $F(-\sqrt{2}, 0), M N: x=n y-\sqrt{2}$,

Solving the system of equations $\left\{\begin{array}{l}x=n y-\sqrt{2}, \\ x^{2}-y^{2}=1\end{array}\right.$
$$
\Rightarrow\left(n^{2}-1\right) y^{2}-2 \sqrt{2} n y+1=0 \text {. }
$$

Let $M\left(x_{1}, y_{1}\right), N\left(x_{2}, y_{2}\right)$, then we have
$$
S_{\triangle O M N}=\frac{\sqrt{2}}{2}\left|y_{1}-y_{2}\right|=\sqrt{2} \cdot \sqrt{n^{2}+1} \text {, }
$$

Therefore, when $n=0$, the minimum area of $\triangle O M N$ is $\sqrt{2}$.
(2) Suppose there exists $P(t, 0)$ that satisfies the condition, then $\overrightarrow{P M}=\left(x_{1}-t, y_{1}\right), \overrightarrow{P N}=\left(x_{2}-t, y_{2}\right)$,
$$
\begin{array}{l}
\overrightarrow{P M} \cdot \overrightarrow{P N}=\left(x_{1}-t\right)\left(x_{2}-t\right)+y_{1} y_{2}=\left(n y_{1}-\sqrt{2}-t\right)\left(n y_{2}-\sqrt{2}-t\right)+y_{1} y_{2} \\
=\left(n^{2}+1\right) y_{1} y_{2}-n(\sqrt{2}+t)\left(y_{1}+y_{2}\right)+(\sqrt{2}+t)^{2} \\
=\frac{1}{n^{2}-1}\left[\left(n^{2}+1\right)-2 \sqrt{2} n^{2}(\sqrt{2}+t)+\left(n^{2}-1\right)(\sqrt{2}+t)^{2}\right] \\
\Rightarrow 1-2 \sqrt{2}(\sqrt{2}+t)=-1 \Rightarrow \sqrt{2}(\sqrt{2}+t)=1 \Rightarrow t=-\frac{\sqrt{2}}{2} .
\end{array}
$$

At this point, $\overrightarrow{P M} \cdot \overrightarrow{P N}=-\frac{1}{2}$, so there exists a point $P\left(-\frac{\sqrt{2}}{2}, 0\right)$ such that $\overrightarrow{P M} \cdot \overrightarrow{P N}$ is a constant.