Linear non-autonomous Cauchy problems and evolution semigroups

TL;DR

This paper investigates the existence of propagators for linear non-autonomous evolution equations using evolution semigroup methods, with applications to time-dependent Schrödinger operators with moving point interactions.

Contribution

It introduces a semigroup-based approach to establish propagator existence for non-autonomous evolution problems, including specific results for Schrödinger operators with time-dependent point interactions.

Findings

Established existence of propagators for hyperbolic type evolution equations.

Applied results to Schrödinger operators with moving point interactions in 1D.

Reduced the problem to perturbation of semigroup generators.

Abstract

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schr\"odinger operators with moving point interactions in 1D.