p-adic Ghobber-Jaming Uncertainty Principle
K. Mahesh Krishna
TL;DR
This work establishes a finite-dimensional p-adic Ghobber-Jaming Uncertainty Principle for two orthonormal bases in a p-adic Hilbert space, showing that if the maximum cross-coherence between the bases over certain index sets is strictly less than 1, then any vector cannot be simultaneously concentrated on those index subsets in both bases; a explicit bound involves the reciprocal of 1 minus the coherence. It then extends the result to non-Archimedean Banach spaces via p-adic coordinate functionals and p-adic orthonormal bases, deriving analogous inequalities and highlighting a fundamental distinction between inner-product-based and non-Archimedean norms. The paper also discusses related bounds via projection operators and basis intertwiners, and closes with open questions regarding p-adic entropy-based uncertainty and p-adic analogues of classical inequality improvements. Overall, it provides the first p-adic and non-Archimedean counterparts to the Ghobber-Jaming uncertainty framework, enriching finite-dimensional harmonic analysis in non-Archimedean settings.
Abstract
Let $\{τ_j\}_{j=1}^n$ and $\{ω_k\}_{k=1}^n$ be two orthonormal bases for a finite dimensional p-adic Hilbert space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} \displaystyle \max_{j \in M, k \in N}|\langle τ_j, ω_k \rangle|<1, \end{align*} where $o(M)$ is the cardinality of $M$. Then for all $x \in \mathcal{X}$, we show that \begin{align} (1) \quad \quad \quad \quad \|x\|\leq \left(\frac{1}{1-\displaystyle \max_{j \in M, k \in N}|\langle τ_j, ω_k \rangle|}\right)\max\left\{\displaystyle \max_{j \in M^c}|\langle x, τ_j\rangle |, \displaystyle \max_{k \in N^c}|\langle x, ω_k\rangle |\right\}. \end{align} We call Inequality (1) as \textbf{p-adic Ghobber-Jaming Uncertainty Principle}. Inequality (1) is the p-adic version of uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}. We also derive analogues of Inequality (1) for non-Archimedean Banach spaces.