Intrinsic ultracontractivity of a Schrödinger semigroup in RNRN

We give a (possibly sharp) sufficient condition on the electric potential q:RN→[0,∞) in the Schrödinger operator A=−Δ+q(x)•A=−Δ+q(x)• on L2(RN)L2(RN) that guarantees that the Schrödinger heat semigroup {e−At:t⩾0} on L2(RN)L2(RN) generated by −A−A is intrinsically ultracontractive  . Moreover, if q(x)≡q(|x|)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)−1/2dr<∞for somer0∈(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.