p-adic Heisenberg-Robertson-Schrodinger and p-adic Maccone-Pati Uncertainty Principles

Abstract

Let $\mathcal{X}$ be a p-adic Hilbert space over a conjugated non-Archimedean valued field $\mathbb{K}$ with $2\ne 0$. Let $A:\mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B:\mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded self-adjoint linear operators. For $x\in \mathcal{D}(A)$ with $⟨x,x⟩=1$, define ${\Delta }_x(A):=\Vert Ax-⟨Ax,x⟩x\Vert .$ Then for all $x\in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $⟨x,x⟩=1$, we show that \begin{align*} (1) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.

6 Pages, 0 Figures. Abstract and Theorem 2.5 corrected