Intrinsic ultracontractivity of a Schrödinger semigroup in RN ✩

We give a (possibly sharp) sufficient condition on the electric potential q :RN → [0,∞) in the Schrödinger operator A=− + q(x)• on L2(RN) that guarantees that the Schrödinger heat semigroup {e−At : t 0} on L2(RN) generated by −A is intrinsically ultracontractive. Moreover, if q(x) ≡ q(|x|) is radially symmetric, we show that our condition on q is also necessary (i.e., truly sharp); it reads ∞ r0 q(r)−1/2 dr <∞ for some r0 ∈ (0,∞). Our proofs make essential use of techniques based on a logarithmic Sobolev inequality, Rosen’s inequality (proved via a new Fenchel–Young inequality), and a very precise asymptotic formula due to HARTMAN and WINTNER. © 2009 Elsevier Inc. All rights reserved