  We show that the spectrum of normalizable states on a Euclidean SL(2,R)/U(1)
black hole exhibits a duality between oscillator states and wound strings. This
duality generalizes the FZZ correspondence, which can be thought of as an
identification between a normalizable mode of dilaton gravity and a mode of the
tachyon with winding number one around the Euclidean time circle. It implies
that normalizable states on a large Euclidean black hole have support at widely
separated scales. In particular, localized states that are extended over the
cap of the cigar (the Euclidian analog of the black hole atmosphere) have a
component that is localized near the tip of the cigar (the analog of the
stretched horizon). As a consequence of this duality, the states exhibit a
transition as a function of radial excitation level. From the perspective of a
low energy probe, low lying states are naturally thought of as oscillator
states in the black hole atmosphere, while at large excitation level they are
naturally described as wound strings. As the excitation level increases, the
size of the states first decreases and then increases. This behavior is
expected to be a general feature of black hole horizons in string theory.
