Related papers: Spectral theory of $p$-adic unitary operator

Spectral theory of $p$-adic unitary operator

URL: http://arxiv.org/abs/2310.12266v2
Date: Wed, 1 Nov 2023 18:28:10 GMT
Title: Spectral theory of $p$-adic unitary operator
Authors: Zhao Tianhong
Abstract summary: The spectrum decomposition of $U$ is complete when $psi$ is a $p$-adic wave function. The abelian extension theory of $mathbbQ_p$ is connected to the topological properties of the $p$-adic unitary operator.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
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.

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