Schrödinger semigroups on the scale of Sobolev spaces

Barry Simon

Vol. 122 (1986), No. 2, 475–480

DOI: 10.2140/pjm.1986.122.475

Abstract

We consider the action of semigroups e^−tH, with H = −Δ + V on L²(R^ν), on the scale of Sobolev spaces ℋ^α. We show that while e^−tH maps L² = ℋ⁰ to ℋ² 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 α > 2.

Mathematical Subject Classification 2000

Primary: 47F05

Secondary: 35J10, 35P05, 81C10, 47D05

Milestones

Received: 17 December 1984

Revised: 26 March 1985

Published: 1 April 1986