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An inverse problem for the space-time fractional Schrödinger equation on closed manifolds

Li Li

Abstract

We formulate an inverse problem for an uncoupled space-time fractional Schrödinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the knowledge of the associated source-to-solution map. Our argument relies on the asymptotic behavior of Mittag-Leffler functions, Weyl's law for the eigenvalues of the Laplace-Beltrami operator, the unique continuation property of the space-fractional operator and the disjoint data metric determination result for the wave equation. We also provide a probabilistic formulation of our inverse problem.

An inverse problem for the space-time fractional Schrödinger equation on closed manifolds

Abstract

We formulate an inverse problem for an uncoupled space-time fractional Schrödinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the knowledge of the associated source-to-solution map. Our argument relies on the asymptotic behavior of Mittag-Leffler functions, Weyl's law for the eigenvalues of the Laplace-Beltrami operator, the unique continuation property of the space-fractional operator and the disjoint data metric determination result for the wave equation. We also provide a probabilistic formulation of our inverse problem.

Paper Structure

This paper contains 11 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Connection with earlier literature
  3. Organization
  4. Preliminaries
  5. Laplace-Beltrami operator and Sobolev spaces
  6. Caputo derivatives and Mittag–Leffler functions
  7. Well-posedness
  8. Inverse problem
  9. Determination of the time-fractional power
  10. Determination of the space-fractional power and the metric
  11. Probabilistic formulation

Key Result

Theorem 1.1

Suppose $(M, g)$, $(M, \tilde{g})$ are smooth connected closed Riemannian manifolds. Suppose $0< \alpha, \tilde{\alpha}< 1$ and $0< \beta, \tilde{\beta}< 1$. Let $W_1, W_2\subset M$ be nonempty disjoint open subsets. Suppose $W_1$ satisfies the spectral bound condition (see (specbdcond) below in Su Then $\alpha= \tilde{\alpha}$, $\beta= \tilde{\beta}$, and $(M, g)$ and $(M, \tilde{g})$ are isome

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3: lassas2023disjoint
  • ...and 4 more