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 \( \langle x, x \rangle =1 \), define \( \Delta _x(A):= \|Ax- \langle Ax, x \rangle x \|. \) Then for all \( x \in \mathcal{D}(AB)\cap \mathcal{D}(BA) \) with \( \langle x, x \rangle =1 \), we show that (1)max{Δx(A),Δx(B)}≥|⟨[A,B]x,x⟩2+(⟨{A,B}x,x⟩−2⟨Ax,x⟩⟨Bx,x⟩)2||2| and (2)max{Δx(A),Δx(B)}≥|⟨(A+B)x,y⟩|,∀y∈X satisfying ‖y‖≤1,⟨x,y⟩=0. We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.