Does a quantum particle know the time?

Abstract

We study the spatial regularity of the fundamental solution E(t,x) of the Schrödinger 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.