arXiv subjects
Publications and source records attributed to Anthony Graves-McCleary.
We give matching upper and lower bounds for the Dirichlet heat kernel of a Schrödinger operator $Δ+W$ in the domain above the graph of a bounded Lipschitz function, in the case when $W$ decays away from the boundary faster than quadratically.
We give estimates for positive solutions for the Schrödinger equation $(Δ_μ+W)u=0$ on a wide class of parabolic weighted manifolds $(M, dμ)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan to give matching upper and lower bounds for the heat kernel of the corresponding Schrödinger operator $Δ_μ+W$. In particular, this appears to complement known results for Schrödinger operators on $\mathbf{R}^2$.
We prove a generalized version of the $3G$ Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher-dimensional fractals such as Sierpinski carpets in $\mathbf{R}^n$, $n\geq 3$, as well as generalized fractal-type spaces that do not have a well-defined Hausdorff dimension or walk dimension. This yields new instances of the $3G$ Principle for these spaces. We also discuss applications to Schrödinger operators.