Spectral theory of $p$-adic unitary operator

The $p$-adic unitary operator $U$ is defined as an invertible operator on $p$-adic ultrametric Banach space such that $\mid U\mid =\mid U^{-1}\mid =1$. We point out $U$ has a spectral measure valued in $\mathbf{\text{projection}}\mathbf{\text{functors}}$, 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 $\mathbf{\text{the}}\mathbf{\text{Galois}}\mathbf{\text{theory}}\mathbf{\text{of}}\mathbf{\text{operators}}$. 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: $\mathbf{\text{Teichm}}\ddot{\mathbf{\text{u}}}\mathbf{\text{ller}}\mathbf{\text{type}},\mathbf{\text{continuous}}\mathbf{\text{type}},\mathbf{\text{pro-finite}}\mathbf{\text{type}}$. Finally, we establish a \textbf{framework ofp-adic quantum mechanics}, where projection functor plays a role of quantum measurement.