Lev Kapitanski, Igor Rodnianski
Does a quantum particle know the time?
(43K, AMSTeX)
ABSTRACT. 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.