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Heisenberg Uncertainty Principle on half spaces and Orthants: Best constants, Optimizers and Stability

Nguyen Lam, Yukta Lodha, Guozhen Lu, Ambar N. Sengupta

Abstract

Though the sharp Heisenberg Uncertainty Principle has been extensively studied in the entire Euclidean spaces, the counterpart on the half spaces or more general orthants has been missing in the literature. We investigate the sharp Heisenberg Uncertainty Principle on orthants by computing explicitly the optimal constant and determining all possible extremal functions. Moreover, we establish several stability estimates of the Heisenberg Uncertainty Principle on the half spaces and orthants.

Heisenberg Uncertainty Principle on half spaces and Orthants: Best constants, Optimizers and Stability

Abstract

Though the sharp Heisenberg Uncertainty Principle has been extensively studied in the entire Euclidean spaces, the counterpart on the half spaces or more general orthants has been missing in the literature. We investigate the sharp Heisenberg Uncertainty Principle on orthants by computing explicitly the optimal constant and determining all possible extremal functions. Moreover, we establish several stability estimates of the Heisenberg Uncertainty Principle on the half spaces and orthants.
Paper Structure (6 sections, 22 theorems, 171 equations)

This paper contains 6 sections, 22 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Integration formulas on Orthants
  3. Functions that vanish on the orthant walls
  4. The Heisenberg Uncertainty Principle on an orthant: Proof of Theorem \ref{['T:HeisOrth']}
  5. The Gaussian Poincaré Inequality with monomial weights
  6. Stability of the Heisenberg Uncertainty Principle on an Orthant-Proof of Theorem \ref{['T2']}

Key Result

Theorem 1.1

Let $n \geq 1$, and $k\in\{1,\ldots, n\}$. Then for all $u \in S({{\mathbb R}^n_{k,+}})$. The constant $\frac{(n+2k)^2}{4}$ in (E:HUPorthant) is sharp, and equality holds in (E:HUPorthant) if and only if $u(x)$ is of the form $( \prod_{i=n-k+1}^{n}x_i )\alpha e^{-\beta|x|^2}$, for some constants $\alpha$ and $\beta$, with $\beta>0$.

Theorems & Definitions (39)

  • Theorem 1.1: HUP on orthants
  • Theorem 1.2: HUP on half spaces
  • Theorem A
  • Theorem B
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 29 more