The Schrödinger Equation on Spheres

is a distribution on R×Sd with fairly nasty behavior; its singular support is all of R×Sd. However, J. Rauch pointed out to me that when d = 1 and t is a rational multiple of π (we say t ∈ πQ), then e−it∆δ(x) ∈ D′(S1) is a finite sum of delta functions on S1. Hence for such t ∈ πQ, e−it∆ is bounded on Lp(S1) for each p ∈ [1,∞]. Here we work out an equally precise description of e−it∆ on D′(Sd), for each t ∈ πQ. From this follows a precise account of the Lp-Sobolev mapping properties of e−it∆, for such t. In §2 we will derive the basic identities for e−it∆ on D′(Sd) when t ∈ πQ. For such t we express e−it∆ in terms of solution operators to a wave equation. This leads to the sharp Lp-Sobolev estimates. Establishing sharpness is simply a matter of showing that certain coefficients in the formula for e−it∆ do not vanish. This issue is settled in §3. In §4 we discuss various extensions of these results. It is mentioned that another extension of the S1 case is to d-dimensional tori, and that formulas there have a number-theoretical significance. We also discuss extensions to Zoll surfaces and to situations where a potential is added to the Laplace operator.