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One of the crucial properties of a quantum system is the existence of boundstates. While the existence of eigenvalues below zero, i.e., below theessential spectrum, is well understood, the situation of zero energy boundstates at the edge of the essential spectrum is far less understood. We presentnecessary and sufficient conditions for Schr\"odinger operators to have a zeroenergy bound state. Our sharp criteria show that the existence andnon-existence of zero energy ground states depends strongly on the dimensionand the asymptotic behavior of the potential. There is a spectral phasetransition with dimension four being critical.

Quantum Systems at the Brink: Existence of Bound States, Critical Potentials and Dimensionality