# Eigenvalues of the $p$-adic Harmonic oscillator?

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.

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There is a school of (mostly Russian) theoretical physicists who work on such things -- look up the papers of E.I. Zelenov and B.G. Dragovich, among others. Frankly, despite being a number theorist working on p-adic automorphic forms, I can't make head or tail of any of it, but perhaps it will mean something to you... –  David Loeffler Oct 16 '11 at 9:04
What exactly are $D_q$ and $x_q$? What kind of function is $f$ and what kind of variable is $x$? –  Qiaochu Yuan Mar 5 '12 at 3:40
$D_{q}$ is the q-adic definition of the derivative for finite prime 'q' $x_{q}$ is the position of the harmonic oscillator taking only values on $Q_{q}$ p-adic –  Jose Garcia Mar 5 '12 at 12:43
Can explain the q-adic definition of the derivative for finite prime a little more? Maybe you show, what you have done so far... –  draks ... Apr 19 '12 at 12:32