Forward and backward problems for abstract time-fractional Schrödinger equations

TL;DR

This work establishes forward and backward well-posedness for abstract time-fractional Schrödinger equations on a Hilbert space, distinguishing two operator models: $i\partial_t^\alpha u + Au=0$ with $\nu=1$ and $i^\alpha \partial_t^\alpha u + Au=0$ with $\nu=\alpha$, across subdiffusive ($0<\alpha<1$) and (where applicable) superdiffusive ($(1<\alpha<2)$) regimes. Central to the analysis are eigenfunction expansions and Mittag-Leffler kernels, with detailed study of zeros on the imaginary axis and asymptotics that enable explicit solution formulas and stability estimates. The paper proves forward well-posedness in both models, derives backward uniqueness and stability results that depend on $\alpha$ and time horizon $T$, and identifies discrete critical time sets where backward reconstruction may fail or require extra conditions. It also outlines open problems, including conjectures about the absence of imaginary-axis zeros for certain ranges of $\alpha$ and the potential for logarithmic convexity-type estimates, which would broaden the applicability of backward methods in fractional quantum dynamics. Overall, the results advance the mathematical understanding of inverse-type backward problems for fractional Schrödinger dynamics and pave the way for future generalizations and applications.

Abstract

We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations $\mathrm{i}^ν\partial_t^αu(t) + A u(t)=0$, $α\in (0,1)\cup (1,2)$ and $ν\in\{1,α\}$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. This kind of equation, which incorporates the Caputo time-fractional derivative of order $α$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($ν=1,\,$ $α\in (0,1)$) and ($ν=α,\,$ $α\in (0,1)\cup (1,2)$). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $ν=1$ and $ν=α$. Our approach employs the solution's eigenvector expansion along with the properties of the Mittag-Leffler functions, including the distribution of zeros and asymptotic expansions. Finally, we conclude with a discussion of some open problems.

Figures (5)

• Figure 1: $|E_{\alpha,1}(\mathrm{i}t)|$ for $\alpha\in \{0.61,0.7,0.8,0.95\}$.
• Figure 2: $E_{2\alpha,1}(-t^2)$ and $t E_{2\alpha,1+\alpha}(-t^2)$ for $\alpha=0.8$.
• Figure 3: $|\psi(t)|$ for $\alpha\in \{1.2,1.3,1.4,1.5,1.6,1.7\}$.
• Figure 4: $\mathrm{Re}\, \psi(t)$ and $\mathrm{Im}\, \psi(t)$ for $\alpha=1.1$.
• Figure 5: First derivatives of $f(t)$ for $\alpha=0.35$.

Theorems & Definitions (32)

• Lemma 2.1: see Sc96
• Lemma 2.2: Pages 32-35 in podlubny99
• Corollary 2.3
• Lemma 2.4
• Proposition 2.5
• Remark 1
• Proposition 2.6
• Lemma 3.1
• Definition 3.2
• Theorem 3.3