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We give the definition of $p-adic$ Hermite operator and set up the $p-adic$
spectral measure. We compare the Archimedean case with non-Archimedean case.
The structure of Hermite conjugate in $C^{*}$-Algebra corresponds to three
canonical structures of $p-adic$ ultrametric Banach algebra: 1. mod $p$
reduction 2. Frobenius map 3. Teichm\”uller lift. There is a nature connection
between Galois theory and Hermite operator spectral decomposition. The Galois
group $\mathrm{Gal}(\bar{\mathbb{F}}_p|\mathbb{F}_p)$ generate the $p-adic$
spectral measure. We point out some relationships with $p-adic$ quantum
mechanics: 1. creation operator and annihilation operator 2. $p-adic$
uncertainty principle.

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