A note on interpolation of semigroups

Abstract

Abstract. In this note, we extend the theory of interpolation of semigroups considered by Arendt (1991) and Arendt-Neubrander-Schlotterbeck (1992) to include the fractional order case. We then apply the results to the Schrödinger and wave equations on LP{Q) where Í2 C R^ is open with bounded Lebesgue measure.

On the Schr�dinger equation inL p spaces

“…Using the method of sobolev imbeddings as mentioned in the introduction, one obtains the following (see [14]), see also the proof in the Appendix which is based on the methods of the present paper. The aim of this section is to prove the optimality of this result.…”

“…For the precise definition of A which uses the theory of quadratic forms, see e.g. It follows from [1], [10] and [14] that iA is the generator of an ~-times integrated group for~_~ 89189 1] or Arendt [1].…”

“…This result was later extended to ~ ~ R + in [14]. In [10], trace theorems for holomorphic semigroups were established and applied to the Schrtdinger equation in LP(fl) without the restriction that [2 be of finite measure.…”