We discuss a recently discovered connection between the discrete spectrum and the essential spectrum of Schr"odinger operators in one or two space dimensions. The situation is particularly interesting on the half-line since new phenomena occur in this case due to boundary effects. For example, we show that the existence of singular spectrum embedded in the essential spectrum implies that the discrete spectrum is infinite. The proof starts out by relating the problem at hand to the theory of orthogonal polynomials on the unit circle via the Szeg"o and Geronimus transformation. This transformation yields estimates on the potential, which can then be fed into an analysis of the non-linear Fourier transform arising in the Pr"ufer reformulation of the time-independent Schr"odinger equation.