Inverse problems for time-fractional Schrödinger equations

TL;DR

This work addresses inverse problems for time-fractional Schrödinger equations with Caputo derivatives $\partial_t^\alpha$, $0<\alpha<1$, and a symmetric elliptic operator $\mathcal{L}$. It develops a spectral-analytic framework based on Mittag-Leffler functions, Laplace transforms, and unique continuation to establish refined uniqueness from measurements on sets of positive measure for several problems: inverse initial data, inverse source with separable time factor, and inverse order (recovering $\alpha$). The methods show that low-regularity initial data in $L^2(\Omega)$ and measurements on $E\subset\Omega$ with $|E|>0$ suffice for identifiability, with solutions analytic in a sector $\Sigma$ and explicit spectral representations. The results extend prior work on fractional diffusion to the Schrödinger setting and have implications for controllability and identifiability in fractional quantum-type models.

Abstract

We study some inverse problems for time-fractional Schrödinger equations involving the Caputo derivative of fractional order $α\in (0,1)$. We prove refined uniqueness results from sets of positive Lebesgue measure for various problems by weakening the regularity of initial data.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Proposition 2.2
• Proposition 3.1
• Remark 2