7. Let positive integers $a$, $m$, $n$ satisfy
$$
\sqrt{a^{2}-4 \sqrt{2}}=\sqrt{m}-\sqrt{n} \text {. }
$$

Then the values of $a$, $m$, $n$ ( ).
(A) have one set
(B) have two sets
(C) have more than two sets
(D) do not exist
7. A.

Squaring the given equation, we get
$$
\begin{array}{l}
a^{2}-4 \sqrt{2}=m+n-2 \sqrt{m n} . \\
\text { Then }\left\{\begin{array}{l}
m n=8, \\
a^{2}=m+n .
\end{array}\right.
\end{array}
$$

Given $m \geqslant n$, we know $(m, n)=(8,1)$ or $(4,2)$.
Upon verification, the solution that meets the conditions is
$$
(m, n, a)=(8,1,3) \text {. }
$$

Therefore, the answer is A.