## 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

Zbl 0713.58055
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### References:

 [1] M. Ben-Artzi, Global estimates for the Schrödinger equation , J. Funct. Anal. 107 (1992), no. 2, 362-368. · Zbl 0774.35019 [2] M. Ben-Artzi and A. Devinatz, Local smoothing and convergence properties of Schrödinger type equations , J. Funct. Anal. 101 (1991), no. 2, 231-254. · Zbl 0762.35022 [3] J. Bergh and J. Löfström, Interpolation Spaces. an Introduction , Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin, 1976. · Zbl 0344.46071 [4] P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations , J. Amer. Math. Soc. 1 (1988), no. 2, 413-439. JSTOR: · Zbl 0667.35061 [5] P. Constantin and J. C. Saut, Local smoothing properties of Schrödinger equations , Indiana Univ. Math. J. 38 (1989), no. 3, 791-810. · Zbl 0712.35022 [6] S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions , J. Math. Kyoto Univ. 34 (1994), no. 2, 319-328. · Zbl 0807.35026 [7] M. P. Gaffney, The harmonic operator for exterior differential forms , Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 48-50. JSTOR: · Zbl 0042.10205 [8] N. Hayashi, K. Nakamitsu, and M. Tsutsumi, On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension , Math. Z. 192 (1986), no. 4, 637-650. · Zbl 0617.35025 [9] N. Hayashi, K. Nakamitsu, and M. Tsutsumi, On solutions of the initial value problem for the nonlinear Schrödinger equations , J. Funct. Anal. 71 (1987), no. 2, 218-245. · Zbl 0657.35033 [10] 1 L. Hörmander, The analysis of linear partial differential operators. I , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. · Zbl 0521.35001 [11] 2 L. Hörmander, The analysis of linear partial differential operators. III , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. · Zbl 0601.35001 [12] W. Ichinose, On $$L^ 2$$ well posedness of the Cauchy problem for Schrödinger type equations on the Riemannian manifold and the Maslov theory , Duke Math. J. 56 (1988), no. 3, 549-588. · Zbl 0713.58055 [13] K. Kajitani, The Cauchy problem for Schrödinger type equations with variable coefficients , · Zbl 0917.35129 [14] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation , Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93-128. · Zbl 0549.34001 [15] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect , Rev. Math. Phys. 1 (1989), no. 4, 481-496. · Zbl 0833.47005 [16] H. Kumano-go, Pseudodifferential operators , MIT Press, Cambridge, Mass., 1981. · Zbl 0489.35003 [17] S. Mizohata, Some remarks on the Cauchy problem , J. Math. Kyoto Univ. 1 (1961/1962), 109-127. · Zbl 0104.31903 [18] M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975. · Zbl 0308.47002 [19] P. Sjölin, Regularity of solutions to the Schrödinger equation , Duke Math. J. 55 (1987), no. 3, 699-715. · Zbl 0631.42010 [20] W. Strauss, Smoothing of dispersive waves , Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1993), École Polytech., Palaiseau, 1993, Exp. No. XIV, 5. · Zbl 0818.35098 [21] K. Yajima, On smoothing property of Schrödinger propagators , Functional-analytic methods for partial differential equations (Tokyo, 1989), Lecture Notes in Math., vol. 1450, Springer, Berlin, 1990, pp. 20-35. · Zbl 0725.35084 [22] M. Yamazaki, On the microlocal smoothing effect of dispersive partial differential equations. I. Second-order linear equations , Algebraic analysis, Vol. II, Academic Press, Boston, MA, 1988, pp. 911-926. · Zbl 0683.35010
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