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

Let $\mathcal{X}$ be a p-adic Hilbert space. 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 ${\mathrm{\Delta }}_x(A):=\mathrm{\parallel }Ax-⟨Ax,x⟩x\mathrm{\parallel }\mathrm{.}$ Then for all $x\in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $⟨x,x⟩=1$, we show that \begin{align*} (1) \quad \quad \quad \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 \quad \quad \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.