$p$-Adic and adelic harmonic oscillator with a time-dependent frequency

The classical and quantum formalism for a $p$-adic and adelic harmonic oscillator with a time-dependent frequency is developed, and general formulas are obtained for the main theoretical quantities. In particular, the $p$-adic propagator is calculated, and the existence of a simple vacuum state as well as adelic quantum dynamics is shown. A spatial discreteness and a $p$-adic quantum mechanical phase are noted.