On fundamental solutions of generalized Schrödinger operators
Let \(\mu\) be a nonnegative Radon mesure on \(\mathbb{R}^n\), \(n\geq 3\), and consider the Schrödinger operator. If \(\mu\) satisfies certain scaling invariant Kato conditions and doubling conditions, the following estimates for the fundamental solution \(\Gamma_\mu\) of \(-\Delta+\mu\) are proved: \[ {ce^{-\varepsilon_2 d(x,y;\mu)} \over|x-y|^{u-2}} \leq\Gamma_\mu (x,y)\leq {Ce^{-\varepsilon_1 d(x,y;\mu)} \over|x-y|^{n-2}}, \] where \(d(x,y;\mu)\) is the distance function for the modified Agmon metric associated with \(\mu\). Further the boundedness of the corresponding Riesz transform \(\text{grad}(-\Delta +\mu)^{-1/2}\) is investigated on \(L^p\).
Schrödinger operator with measure as potential
scientific article; zbMATH DE number 1367857
scientific article