Schrödinger semigroups on the scale of Sobolev spaces

We consider the action of semigroups e(-tH), with H = -Δ + V on L(2)(R(v)), on the scale of Sobolev spaces H(α). We show that while e(-tH) maps L(2)=H(0) to H(2) under great generality, there exist bounded V so that, for all β > 0, e(-tH)[H(β)] is not contained in any H(α) with a > 2.