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