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 $\textbf{projection functors}$, 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 $\textbf{the Galois theory of operators}$. 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: $\textbf{Teichm\"uller type}, \textbf{continuous type}, \textbf{pro-finite type}$. Finally, we establish a $\textbf{framework of $p$-adic quantum mechanics}$, where projection functor plays a role of quantum measurement.