11. (This question is worth 20 points) In the Cartesian coordinate system $x O y$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following condition: Draw any two perpendicular lines $l_{1}$ and $l_{2}$ through the point $(a, 0)$. If $l_{1}$ intersects the hyperbola $C$ at points $P, Q$, and $l_{2}$ intersects $C$ at points $R, S$, then $|P Q|=|R S|$ always holds.
Draw two perpendicular lines $l_{1}: x=a$ and $l_{2}: y=0$ through the point $(a, 0)$.
It is easy to see that $l_{1}$ intersects $C$ at points $P_{0}\left(a, \sqrt{a^{2}-1}\right)$ and $Q_{0}\left(a, -\sqrt{a^{2}-1}\right)$ (note that $a > 1$), and $l_{2}$ intersects $C$ at points $R_{0}(1,0)$ and $S_{0}(-1,0)$. From the given conditions, we have $2 \sqrt{a^{2}-1} = |P_{0} Q_{0}| = |R_{0} S_{0}| = 2$, solving this gives $a = \sqrt{2}$.
This means that the only possible value for $a$ is $\sqrt{2}$. $\qquad$
Next, we verify that $a = \sqrt{2}$ satisfies the conditions.
In fact, when one of the lines $l_{1}$ or $l_{2}$ has an undefined slope, we can set $l_{1}: x=a$ and $l_{2}: y=0$, which is the case discussed earlier for $l_{1}$ and $l_{2}$, and in this case, $|P Q| = |R S|$. If both $l_{1}$ and $l_{2}$ have defined slopes, we can assume
$$
l_{1}: y = k(x - \sqrt{2}), \quad l_{2}: y = -\frac{1}{k}(x - \sqrt{2}) \quad (k \neq 0),
$$

Note that $k \neq \pm 1$ (otherwise $l_{1}$ will be parallel to the asymptotes of $C$, and thus $l_{1}$ will intersect $C$ at only one point).
By solving the system of equations for $l_{1}$ and $C$, we get $x^{2} - k^{2}(x - \sqrt{2})^{2} - 1 = 0$, which simplifies to
$$
(1 - k^{2}) x^{2} - 2 \sqrt{2} k^{2} x - 2 k^{2} - 1 = 0,
$$

This is a quadratic equation, and its discriminant is $\Delta = 4 k^{2} + 4 > 0$. Therefore, $l_{1}$ intersects $C$ at two distinct points $P$ and $Q$. Similarly, $l_{2}$ intersects $C$ at two distinct points $R$ and $S$. By the chord length formula, we have
$$
|P Q| = \sqrt{1 + k^{2}} \cdot \frac{\sqrt{4 k^{2} + 4}}{|1 - k^{2}|} = 2 \cdot \left| \frac{1 + k^{2}}{1 - k^{2}} \right|.
$$

By substituting $-\frac{1}{k}$ for $k$, we similarly get $|R S| = 2 \cdot \left| \frac{1 + (-k)^{-2}}{1 - (-k)^{-2}} \right| = 2 \cdot \left| \frac{k^{2} + 1}{k^{2} - 1} \right|$. Thus, $|P Q| = |R S|$. In conclusion, $a = \sqrt{2}$ is the value that satisfies the conditions.
20 points