Evolution equation with fractional Schr\"odinger operators: monotonicity and exponential decay of solutions in Morrey spaces

TL;DR

This paper studies evolution equations involving fractional Schrödinger operators within Morrey spaces, establishing properties of the associated semigroup, including monotonicity, exponential growth estimates, and conditions for decay.

Contribution

It introduces new monotonicity results and decay conditions for semigroups generated by fractional Schrödinger operators in Morrey spaces, extending previous theoretical frameworks.

Findings

Semigroup preserves order in Morrey spaces.

Exponential growth estimates for the semigroup.

Necessary and sufficient conditions for exponential decay of Morrey norms.

Abstract

We consider evolution equation with fractional Schr\"odinger operators in Morrey spaces. We prove order preserving properties of the associated semigroup in Morrey scale. We prove monotonicity of the semigroup with respect to Morrey's potentials and give some precise estimates of its exponential growth. We show that Arendt and Batty's type condition on the potential is necessary for exponential decay of Morrey's norms of the semigroup and find a large class of dissipative potentials for which it is also sufficient.