Spectral theory of $p$-adic unitary operator

Abstract

The $p$-adic unitary operator $U$ is defined as an invertible operator on $p$-adic ultrametric Banach space such that $\left|U\right|=\left|U^{-1}\right|=1$. We point out $U$ has a spectral measure valued in $\text{projection functors}$, which can be explained as the measure theory on the formal group scheme. The spectrum decomposition of $U$ is complete when $\psi$ is a $p$-adic wave function. We study $\text{the Galois theory of operators}$. The abelian extension theory of ${\mathbb{Q}}_p$ is connected to the topological properties of the $p$-adic unitary operator. We classify the $p$-adic unitary operator as three types: $\text{Teichmu¨ller type},\text{continuous type},\text{pro-finite type}$. Finally, we establish a $\textbf{framework of $p$-adic quantum mechanics}$, where projection functor plays a role of quantum measurement.