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Quantitative uniqueness of continuation for the Schrödinger equation : explicit dependence on the potential

Mourad Choulli, Hiroshi Takase

TL;DR

This work delivers a comprehensive quantitative theory of unique continuation for the Schrödinger operator $-\Delta+V$ with potentially unbounded $V$ on bounded Lipschitz domains. By combining sharp Carleman inequalities (including a new one for $V\in L^s$ with $s>n/2$) with Caccioppoli-type estimates and three-ball inequalities, the authors obtain explicit, potential-dependent constants and both interior and global continuation results, along with doubling inequalities and vanishing-order estimates. They extend the framework to first-order perturbations $-\Delta +W\cdot\nabla +V$ and to quantitative uniqueness from Cauchy data on boundaries. The findings advance the understanding of how the size and regularity of the potential control the rate at which solutions can vanish and how interior information propagates to global information, with clear implications for stability in inverse problems and control theory.

Abstract

We demonstrate a quantitative version of the usual properties related to unique continuation from an interior datum for the Schrödinger equation with bounded or unbounded potential. The inequalities we establish have constants that explicitly depend on the potential. We also indicate how the above-mentioned inequalities can be extended to elliptic equations with bounded or unbounded first-order derivatives. The case of unique continuation from Cauchy data is also considered.

Quantitative uniqueness of continuation for the Schrödinger equation : explicit dependence on the potential

TL;DR

This work delivers a comprehensive quantitative theory of unique continuation for the Schrödinger operator with potentially unbounded on bounded Lipschitz domains. By combining sharp Carleman inequalities (including a new one for with ) with Caccioppoli-type estimates and three-ball inequalities, the authors obtain explicit, potential-dependent constants and both interior and global continuation results, along with doubling inequalities and vanishing-order estimates. They extend the framework to first-order perturbations and to quantitative uniqueness from Cauchy data on boundaries. The findings advance the understanding of how the size and regularity of the potential control the rate at which solutions can vanish and how interior information propagates to global information, with clear implications for stability in inverse problems and control theory.

Abstract

We demonstrate a quantitative version of the usual properties related to unique continuation from an interior datum for the Schrödinger equation with bounded or unbounded potential. The inequalities we establish have constants that explicitly depend on the potential. We also indicate how the above-mentioned inequalities can be extended to elliptic equations with bounded or unbounded first-order derivatives. The case of unique continuation from Cauchy data is also considered.

Paper Structure

This paper contains 16 sections, 39 theorems, 378 equations.

Table of Contents

  1. Introduction
  2. Caccioppoli-type inequalities
  3. Three-ball inequalities
  4. Potential in $L^s$, $s\in (n/2,\infty)$
  5. Potential in $L^\infty$
  6. Potential in $L^{n/2}$
  7. Quantitative uniqueness of continuation
  8. Global quantitative uniqueness of continuation
  9. Global quantitative uniqueness of continuation for $C^{1,1}$ domains
  10. Doubling inequality
  11. Quantitative vanishing order
  12. Quantitative vanishing order for continuous potentials
  13. Extensions
  14. Quantitative uniqueness of continuation from Cauchy data
  15. Covering lemma
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $s\in (n/2,\infty)$ and $V\in L^s(\Omega)$. For all $t>0$, $V$ can be decomposed into two terms $V=V_t^1+V_t^2$ so that $V_t^1\in L^{n/2}(\Omega)$, $V_t^2\in L^\infty (\Omega)$ and

Theorems & Definitions (54)

  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Proposition 3.1
  • Lemma 3.1
  • proof
  • ...and 44 more