Intrinsic ultracontractivity of a Schrödinger semigroup in \(\mathbb R^N\)
Let \(\mathcal{A}:=-\Delta +q(x)\cdot\) denote the Schrödinger operator with a potential \(q:\mathbb{R}^N\to \mathbb{R}\) assumed to be continuous, have positive infimum, and tend to infinity as \(|x|\to \infty\). The authors present a general condition on \(q\) for the Schrödinger semigroup \(\{e^{-\mathcal{A}t}; \;t\geq 0\}\) on \(L^2(\mathbb{R}^N)\) generated by \(-\mathcal{A}\) to be intrinsically ultracontractive. The condition is shown to be necessary if \(q\) is radially symmetric. The two main ingredients in their proofs are a logarithmic Sobolev inequality and a new version of Rosen's inequality.
logarithmic Sobolev and Rosen's inequalities
scientific article; zbMATH DE number 5572179
scientific article