Given a prime $q$, what are the values of the $p$-adic Harmonic oscillator that is the solution to the following $p$-adic differential equation?
$ -D^2_q f(x)+ x_q^2 f(x) = E_n f(x) .$
What are the eigenvalues and eigenfunctions? For $q=\infty$, the system is the harmonic oscillator.