2. Given the function $f(x)=a x^{2}-\sqrt{2}$. If $f(f(\sqrt{2}))=-\sqrt{2}$,
then $a=$ $\qquad$
2.0 or $\frac{\sqrt{2}}{2}$.

From $a\left[a(\sqrt{2})^{2}-\sqrt{2}\right]^{2}-\sqrt{2}=-\sqrt{2}$, we get $a(2 a-\sqrt{2})^{2}=0$.
Solving this, we get $a=0$ or $\frac{\sqrt{2}}{2}$.