Schrödinger semigroups on the scale of Sobolev spaces (Q1076937)

The author considers the smoothing properties of Schrödinger semigroups \(e^{-tH}\) with \(H=\Delta +V\) on \(L^ 2(R^{\nu})\), on the scale of Sobolev spaces \({\mathcal H}\), and show that while \(e^{-tH}\) maps \(L^ 2={\mathcal H}^ 0\) to \({\mathcal H}^ 2\), there exist bounded V so that for all \(\beta >0\), \(e^{-tH}[{\mathcal H}]\) is not contained in any \({\mathcal H}^{\alpha}\) with \(\alpha >2\).