  We study the spatial regularity of the fundamental solution E(t,x) of the
Schr\"odinger equation on the circle in a scale of Besov spaces. Although the
fundamental solution is not smooth, we reveal a fine change of regularity of
E(t,x) at different times t. For rational t, E(t,x) is a weighted sum of
delta-functions, and, therefore, exhibits the same regularity as at t=0. For
irrational t, the regularity of E(t,x) is better and depends on how well t is
approximated by rationals. For badly approximated t (e.g., when t is a
quadratic irrational, or, more generally, when t has bounded quotients in its
continued fraction expansion), E(t,x) is a "1/2-derivative" more regular than
E(0,x). For a generic irrational t, E(t,x) is almost "1/2-derivative" more
regular. However, the better t is approximated by rationals, the lower is the
regularity of E(t,x). We describe different thin classes of irrationals which
prescribe their particular regularity to the fundamental solution. These
classes are singled out and characterized by the behavior of the continued
fraction expansions of their members.
