Smoothing effect for Schrödinger evolution equation via commutator algebra. (English) Zbl 1054.58012

Doi, Shin-ichi

Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 1996-1997, Exp. No. XX, 15 p. (1997).

From the introduction: This article concerns smoothing effects of dispersive evolution equations, especially Schrödinger evolution equations associated with complete Riemannian metrics. Our aim is to clarify the relationship between the global behavior of the Hamilton flow of the principal symbol and the smoothing effects specified later. More precisely we consider two problems.
Problem 1: Define for the complete Riemannian metric \(g\) the subset \(S(g)\) of the unit cotangent bundle where a certain microlocal smoothing effect of the associated Schrödinger evolution group does not hold, and describe it as precisely as possible in terms of the geodesic flow.
Problem 2: Prove (higher order) smoothing effects not only for metrics with short range perturbation of the Euclidean metric but also for other types of metrics including (i) metrics with long range perturbation of the Euclidean metric, (ii) conformally compact metrics, (iii) metrics of warped products.

MSC:

58J10	Differential complexes
58J20	Index theory and related fixed-point theorems on manifolds
35J10	Schrödinger operator, Schrödinger equation
47D06	One-parameter semigroups and linear evolution equations
47F05	General theory of partial differential operators