Spectral theory of p-adic Hermite operator

The p-adic Hermite operator is defined and the p-adic spectral measure is set up. Furthermore, a comparison between the Archimedean case and the non-Archimedean case is conducted. The Hermite conjugate in C^*-algebra corresponds to three canonical structures of p-adic ultrametric Banach algebra: mod p reduction, Frobenius map, and Teichmüller lift. There is a connection between Galois theory and Hermite operator spectral decomposition. The Galois group \rmGal(\bar\mathbbF_p|\mathbbF_p) generates the p-adic spectral measure. Finally, some relationships with p-adic quantum mechanics are highlighted: creation operator and annihilation operator, as well as p-adic uncertainty principle.