Spectral theory of $p-adic$ Hermite operator

TL;DR

This paper introduces the $p$-adic Hermite operator, establishes its spectral measure, and explores its connections with Galois theory, $p$-adic quantum mechanics, and the structure of $p$-adic Banach algebras.

Contribution

It defines the $p$-adic Hermite operator, constructs its spectral measure, and links it to Galois groups and $p$-adic quantum mechanics, highlighting novel structural insights.

Findings

The $p$-adic Hermite operator's spectral measure is generated by the Galois group.

Connections between $p$-adic spectral theory and quantum mechanics are established.

Structural parallels between $p$-adic Banach algebras and Hermite conjugates are identified.

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 $C^{*}$-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.