On the images of Sobolev spaces under the Schrödinger semigroup

Abstract

Abstract In this article, we consider the Schrödinger semigroup for the Laplacian Δ on {\mathbb{R}^{n}} , and characterize the image of a Sobolev space in {L^{2}(\mathbb{R}^{n},e^{u^{2}}du)} under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on {\mathbb{R}^{n}} .