Spectral theory of $p-adic$ Hermite operator

Abstract

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 $C^{\ast }$-Algebra corresponds to three canonical structures of $p-adic$ ultrametric Banach algebra: 1. mod $p$ reduction 2. Frobenius map 3. Teichmüller 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.