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Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. (English) Zbl 0870.58101
The author studies conditions under which the Schrödinger group \(\exp (it \Delta)\), where \(\Delta\) is the Laplacian on a complete (connected) manifold \(M\), fails to be microlocally smoothing. The author’s main points are as follows: If \(S(g)\) is the set of all points in the unit tangent bundle at which the group is not smoothing (that is, does not lead to the usual gain of half a derivative), then \(S(g)\) is invariant under the Hamilton flow of the symbol of \(\sqrt {-\Delta}\). If the volume of \(M\) is finite, \(S(g)= S^*M\). If \(S(g)\) is compact, and if no complete geodesic remains in a compact set, then \(S(g)\) is empty. This compactness condition is realized in a number of concrete cases, such as asymptotically flat manifolds. This work therefore complements positive results on local smoothing under ‘non-trapping’ conditions.
The proofs rest on the explicit construction of solutions which violate the smoothing condition. As the author notes, the constructions are similar in spirit to the construction of asymptotic solutions of the Cauchy problem, as by W. Ichinose [Duke Math. J. 56, No. 3, 549-588 (1988; Zbl 0713.58055)]. The details are somewhat lengthy, but the presentation is reasonably clear.

MSC:
58J47 Propagation of singularities; initial value problems on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
35B65 Smoothness and regularity of solutions to PDEs
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