Characterization of intrinsic ultracontractivity for one dimensional Schrödinger operators (Q1727406)

The authors study a Schrödinger operator \(L = - d^2 / d x^2 + V(x)\) on \(\mathbb{R}\), where \(V\) is a real-valued measurable function. Operator \(L\) is said to be \textit{intrinsic ultracontractive} (IU) if self adjoint operator \(L\) has the first positive eigenvalue with positive \(L^2\) eigenfunction \(\phi_0\) and the kernel \(p(x,y,t)\) of the semigroup \(e^{-tL}\) is such \(p(x,y,t) \leq C(t)\phi_0(x) \phi_0(y), x,y\in {\mathbb R}, \forall t\in(0,1]\). The authors consider potentials \(V(x)=q(x)+r(x)\), where \(\lim\limits_{|x|\to\infty}q(x)=\infty\) and \[ q(\cdot)^{(1-3\gamma)/2}|q'(\cdot)|^\gamma \in L^1({\mathbb R}), \gamma\in (1,2], \] \[ q(\cdot)^{-1/2}r(\cdot) \in L^1({\mathbb R}). \] It is proved that \(p\) satisfies (IU) if and only if \(q(\cdot)^{-1/2} \in L^1({\mathbb R})\) and \[ q(x)^{-1}\int\limits_0^x q(s)^{1/2}ds\to 0 \text{ as } |x| \to \infty. \] New examples of potentials for which the semigroups satisfy (IU) or non-(IU) are presented.