  A recent heuristic argument based on basic concepts in spectral analysis
showed that the twin prime conjecture and a few other related primes counting
problems are valid. A rigorous version of the spectral method, and a proof for
the existence of infinitely many quadratic twin primes $n^{2}+1$ and $n^{2}+3$,
$n \geq 1$, are proposed in this note.
