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