Maximal regularity for non-autonomous Schroedinger type equations

TL;DR

This paper establishes maximal regularity results for non-autonomous Schrödinger-type equations, demonstrating regularity under Hölder continuity in Hilbert spaces and $L^p-L^q$ estimates for variable potentials in non-Hilbert spaces.

Contribution

It proves maximal regularity for non-autonomous evolution equations with Hölder continuous operators on Hilbert spaces and extends $L^p-L^q$ estimates to Schrödinger operators with time-dependent potentials in non-Hilbert spaces.

Findings

Maximal regularity holds under Hölder continuity assumptions.

Established $L^p-L^q$ estimates for Schrödinger operators with variable potentials.

Extended regularity results to non-Hilbert space settings.

Abstract

In this paper we study the maximal regularity property for non-autonomous evolution equations $\partial_t u(t)+A(t)u(t)=f(t), u(0)=0.$ If the equation is considered on a Hilbert space $H$ and the operators $A(t)$ are defined by sesquilinear forms $ a(t,.,.)$ we prove the maximal regularity under a Holder continuity assumption of $t \to a(t,.,.)$. In the non-Hilbert space situation we focus on Schrodinger type operators $A(t):= -\Delta + m(t, .)$ and prove $L^p-L^q$ estimates for a wide class of time and space dependent potentials $m$.