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.