In the paper by Keller and Vakil titled "$\pi_p$, the value of $\pi$ in $\ell_p$" (link), it is shown that $\pi_p = 4 \int_{0}^{2^{-1/p}}\left (1+|x^{-p}-1|^{1-p}\right )^{1/p} dx$ and that for $p,q$ such that $\frac{1}{p}+\frac{1}{q}=1$ we have $\pi_p=\pi_q$.
Now consider their graph of $\pi_p$ as a function of $p$ defined on $[1,+\infty)$ :
One could view it as a potential in a 1D Schrödinger equation $-\frac{d^2\psi}{dp^2}(p)+\pi_p\psi(p)=E\psi(p)$ that has bound states, and thus ask :
What is the spectrum of $\pi_p$ ? In particular, can the eigenvalues and eigenfunctions be computed in closed form ? Or do the first few numerical eigenvalues give a hint as to what the result might be ?