Non-Archimedean Heisenberg-Robertson-Schrodinger and Non-Archimedean Maccone-Pati Uncertainty Principles

Let X be a non-Archimedean Hilbert space over a conjugated non-Archimedean valued field K with Char(K) ̸= 2. Let A : D(A) ⊆ X → X and B : D(B) ⊆ X → X be possibly unbounded self-adjoint linear operators. For x ∈ D(A) with ⟨x, x⟩ = 1, define Δx(A) := ∥Ax − ⟨Ax, x⟩x∥. Then for all x ∈ D(AB) ∩ D(BA) with ⟨x, x⟩ = 1, we show that (1) \( \quad \Delta_x(A)+\Delta_x(B)\geq \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \) and (2) \( \quad \Delta_x(A)+\Delta_x(B)\geq \max\{\Delta_x(A), \Delta_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \) Inequality (1) may be regarded as a non-Archimedean analogue of the Heisenberg-Robertson-Schrodinger uncertainty principle, while Inequality (2) serves as a non-Archimedean analogue of the Maccone-Pati uncertainty principle.