Various variants of the matrix
$$A_{J,K} = |J\cap K|$$
were studied, and the spectrum was computed. A one-parameter variant is given by
$$A^{(i)}_{J,K} = \binom{|J\cap K|}{i}$$
for some fixed $i$, and your problem corresponds to $i=1$. In total, six variants are given in a [this friendly paper][1] by Ghareghani, Ghorbani and Mohammad‐Noori.  In particular, the spectrum of $A^{(i)}$ is described in Lemma 9. The lemma is attributed to [R. M. Wilson (1982)][2], who uses somewhat different notation and terminology.

Other useful references, including to additional works of R. M. Wilson, appear in the bibliography of the G-G-MN paper.


  [1]: https://onlinelibrary.wiley.com/doi/full/10.1002/jcd.21308
  [2]: https://www.sciencedirect.com/science/article/pii/0024379582900271