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Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary andlet $K\subset\partial\Omega$ be either a $C^2$ submanifold of the boundary ofcodimension $k<N$ or a point. In this article we study various problems relatedto the Schr\"odinger operator $L_{\mu} =-\Delta - \mu d_K^{-2}$ where $d_K$denotes the distance to $K$ and $\mu\leq k^2/4$. We establish parabolicboundary Harnack inequalities as well as related two-sided heat kernel andGreen function estimates. We construct the associated Martin kernel and proveexistence and uniqueness for the corresponding boundary value problem with datagiven by measures. Next we apply the results to the study of $L_\mu u+g(u) = 0$and establish existence and uniqueness under suitable assumptions on thefunction $g$. To prove our results we introduce among other things a suitablenotion of boundary trace. This trace is different from the one used by Marcusand Nguyen \cite{MT} thus allowing us to cover the whole range $\mu\leq k^2/4$.

Heat and Martin kernel estimates for Schr\"{o}dinger operators with critical Hardy potentials