Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles

Abstract

In this work the ring of finite adèles ${\mathbb {A}}_f$ of the rational numbers ${\mathbb {Q}}$ is obtained as a completion of ${\mathbb {Q}}$ with respect to a certain non-Archimedean metric related to the second Chebyshev function, which allows us to represent any finite adèle as a convergent series, generalizing m-adic analysis. This polyadic analysis allows us to introduce a novel pseudodifferential operator $D^{\alpha }$ on $L^2({\mathbb {A}}_f)$ of fractional differentiation, similar to the Vladimirov operator on the p-adic numbers. The operator $D^{\alpha }$ is a positive selfadjoint unbounded operator whose spectrum $\sigma (D^{\alpha })$ is essential and it consists of a countable number of eigenvalues, which converges to zero, and zero. Moreover, a sort of multiresolution analysis on ${\mathbb {A}}_f$ provides us with a wavelet basis which is an orthonormal basis of eigenfunctions of $D^{\alpha }$ as well. The Cauchy problem of a wave-type pseudodifferential equation

$$\begin{aligned} u_{tt}(x,t)+D^{\alpha }_x u(x,t) =F(x,t), \qquad (x \in {\mathbb {A}}_f), \end{aligned}$$

with appropriate initial conditions $u(x,0)=f(x), u_t(x,0)=g(x),$ and external force F(x, t), is solved separating variables and using the Fourier expansion of functions in $L^2({\mathbb {A}}_f)$, with respect to the wavelet basis.