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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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    Can explain *the q-adic definition of the derivative for finite prime* a little more? Maybe you show, what you have done so far...2012-04-19

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