Optimal Hardy inequalities for Schrödinger operators on graphs

Abstract

For a given subcritical discrete Schrödinger operator $H$ on a weighted infinite graph $X$, we construct a Hardy-weight $w$ which is optimal in the following sense. The operator $H-\lambda w$ is subcritical in $X$ for all $\lambda <1$, null-critical in $X$ for $\lambda =1$, and supercritical near any neighborhood of infinity in $X$ for any $\lambda >1$. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs.

The previous version has been split, the present version is a revision of the second part of the previous one. The first part of the previous version will be posted to arXiv as an independent paper with title "Criticality theory for Schrödinger operators on graphs". 25 pages, 1 figure. Comments are welcome!