Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inverse problem for fractional Schrödinger equations with drift on closed Riemannian manifolds

Tianyu Cai, Xi Chen

TL;DR

This work addresses the inverse problem for a variable-coefficient fractional Schrödinger operator with drift on a closed Riemannian manifold, for order $\alpha\in\left(\tfrac{1}{2},1\right)$. By relating the local source-to-solution map to heat-semigroup and spectral data, the authors show that the metric $g$, drift $b$, and potential $V$ are uniquely determined up to gauge from measurements on any nonempty open set $O$, under suitable geometric conditions. The key technical contributions are a Runge approximation on manifolds to recover the drift, and a gauge-invariant framework that allows simultaneous recovery of lower-order terms and geometry from exterior-type data. The results extend nonlocal Calderón-type inversion to fractional Schrödinger problems on manifolds and provide a principled route from exterior data to full geometric and operator coefficients, with potential implications for geometric inverse problems and nonlocal imaging on manifolds.

Abstract

This paper is concerned about the inverse coefficient problems of variable-coefficient fractional Schrödinger equations with drift on connected closed Riemannian manifolds. We prove that the knowledge of the underlying equation of order $α\in (\frac{1}{2},1)$ on any non-empty open subset of the underlying manifold determines the Riemannian metric, the drift and the potential, simultaneously and uniquely, up to a gauge transformation, under the same geometric assumptions on the observation set as in \cite{feizmohammadi2024calderonproblemfractionalschrodinger}. The method of proof is based on that of \cite{feizmohammadi2024calderonproblemfractionalschrodinger} for fractional Schrödinger operators, with the incorporation of the Runge approximation to recover the drift term.

Inverse problem for fractional Schrödinger equations with drift on closed Riemannian manifolds

TL;DR

This work addresses the inverse problem for a variable-coefficient fractional Schrödinger operator with drift on a closed Riemannian manifold, for order . By relating the local source-to-solution map to heat-semigroup and spectral data, the authors show that the metric , drift , and potential are uniquely determined up to gauge from measurements on any nonempty open set , under suitable geometric conditions. The key technical contributions are a Runge approximation on manifolds to recover the drift, and a gauge-invariant framework that allows simultaneous recovery of lower-order terms and geometry from exterior-type data. The results extend nonlocal Calderón-type inversion to fractional Schrödinger problems on manifolds and provide a principled route from exterior data to full geometric and operator coefficients, with potential implications for geometric inverse problems and nonlocal imaging on manifolds.

Abstract

This paper is concerned about the inverse coefficient problems of variable-coefficient fractional Schrödinger equations with drift on connected closed Riemannian manifolds. We prove that the knowledge of the underlying equation of order on any non-empty open subset of the underlying manifold determines the Riemannian metric, the drift and the potential, simultaneously and uniquely, up to a gauge transformation, under the same geometric assumptions on the observation set as in \cite{feizmohammadi2024calderonproblemfractionalschrodinger}. The method of proof is based on that of \cite{feizmohammadi2024calderonproblemfractionalschrodinger} for fractional Schrödinger operators, with the incorporation of the Runge approximation to recover the drift term.

Paper Structure

This paper contains 13 sections, 19 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The fractional Laplacian
  4. The fractional Sobolev space
  5. The source function set
  6. The heat kernel
  7. Inverse spectral problems
  8. The source-to-solution map
  9. Runge approximation
  10. Recovery of the lower order terms
  11. Recovery of the Riemannian metric
  12. From the source-to-solution map to the heat semigroup
  13. Recovery of the spectral data

Key Result

Theorem 1.1

Suppose $\left(M, g\right)$ is a connected closed Riemannian $n$-manifold, and $O\subset M$ is a nonempty open subset. Consider covector fields $b_1, b_2 \in C^\infty\left(M, \mathbb C \otimes T^\ast M\right)$ and potential functions $V_1, V_2 \in C^\infty\left(M, \mathbb C\right)$ which all vanish

Figures (1)

  • Figure 1: Condition \ref{['eqn : FKU condition']}

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Remark 2
  • Lemma 2.1
  • proof
  • Lemma 2.2: Li--Yau estimates
  • Lemma 2.3: Davies--Gaffney inequalities
  • Lemma 2.4
  • ...and 24 more