Maximal regularity for time-fractional Schr\"odinger equations and application to nonlinear equations

TL;DR

This paper establishes maximal regularity results for time-fractional Schrödinger equations with fractional derivatives, enabling the analysis of well-posedness for related nonlinear equations, using novel techniques avoiding complete monotonicity assumptions.

Contribution

It proves new maximal regularity results for time-fractional Schrödinger equations, including $L^2$ and $L^p$ cases, and applies these to nonlinear equation well-posedness, advancing the mathematical understanding of fractional quantum dynamics.

Findings

Proved maximal $L^2$-regularity using Mittag-Leffler functions without monotonicity assumptions.

Established maximal $L^p$-regularity via operator-valued Mikhlin's multiplier theorem.

Applied regularity results to demonstrate local well-posedness of nonlinear fractional Schrödinger equations.

Abstract

We study the maximal regularity problem for abstract time-fractional Schr\"odinger equations $\partial_t^\alpha(u-u_0) -\mathrm{i} A u=f$, with a fractional derivative $\partial_t^\alpha$ of order $\alpha \in (0,1)$. We assume that $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. First, we prove the maximal $L^2$-regularity by leveraging properties of Mittag-Leffler functions with an imaginary argument. Compared to existing results for the subdiffusion equations, our proof avoids using the complete monotonicity of Mittag-Leffler functions, which seems difficult to prove within the setting of an imaginary argument. Then, we prove the maximal $L^p$-regularity for $p\in (1,\infty)$ using the operator-valued version of Mikhlin's multiplier theorem. Finally, we apply the maximal regularity results to prove the local well-posedness of quasilinear and semilinear time-fractional Schr\"odinger equations.