Linear non-autonomous cauchy problems and evolution semigroups

Abstract

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomousévolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-calledévolution semigroup approach which reduces theéxistence problem for propagators to a perturbation problem of semigroup generators. The results are specied to abstract linear non-autonomousévolutionéquations 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 Schrödinger operators with moving point interactions in 1D.