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On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odingeroperators -\Delta -V, with V being either the critical borderline potentialV(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, underDirichlet boundary conditions. In this work we obtain sharp two-sided estimateson the corresponding heat kernels. To this end we transform the Scr\"odingeroperators into suitable degenerate operators, for which we prove a newparabolic Harnack inequality up to the boundary. To derive the Harnackinequality we have established a serier of new inequalities such as improvedHardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As abyproduct of our technique we are able to answer positively to a conjecture ofE.B.Davies.Comment: 40 page

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains