p-ADIC AND ADELIC HARMONIC OSCILLATOR WITH TIME-DEPENDENT FREQUENCY

In quantum-mechanical experiments, as well as in all measurements, numerical results belong to the field of rational numbers Q. In principle, the corresponding theoretical models could be made using only Q, but it would missed usual effectiveness and beauty of mathematical analysis. So, instead of Q one traditionally applies the field of real numbers R in classical mechanics and the field of complex numbers C in quantum mechanics. R is completion of Q with respect to the metric induced by the absolute value and C is an algebraic extension of R. In addition to R there exist the fields of p-adic numbers Qp as completions of Q with respect to p-adic norms (p= a prime number) [1]. According to the Ostrowski theorem, R and Qp (for every p) exhaust all possible completions of Q. Thus Q is dense not only in R but also in each Qp. Therefore, in the last decade there has been a lot and successful interest in construction of theoretical models with p-adic numbers (for a review, see, Refs. 2-5).