Image characterization of certain Sobolev spaces under Schr¨odinger semigroup

This thesis deals with the study of characterizing the image of some function spaces under Schr¨odinger semigroup. This thesis is divided in to four parts. In chapter 2 we consider the Schr¨odinger semigroup for Laplacian � on Rn and characterize the image of Sobolev space on Rn under this semi group as weighted Bergman space HL2(C; um t (z)dz) (up to equivalence of norms). Similarly we characterize the image of Hermite-Sobolev space under Schr¨odinger semigroup associated to Hermite operator H on Rn. In chapter 3, we consider the Schr¨odinger semigroup for the Dunkl-Laplacian ��, associated to a finite reflection group G on Rn. We characterize the image of L2(Rn; eu2h�(u)du) under this semigroup as a reproducing kernel Hilbert space. For the reflection group Zn 2 , we identify the images of the Dunkl-Sobolev spaces in L2(Rn; h�(u)du) under the associated Schr¨odinger semigroup eit� as a reproducing kernel Hilbert space (up to equivalence of norms). Also we establish similar kind of results for Schr¨odinger semigroup associated to Dunkl-Hermite operator. Chapter 4 deals with the sampling in the space, HL2(C; um t (z)dz). We show that for given a separated sequence Z in C is a sampling for HL2(C; u0 t (z)dz) if and only if its lower density