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Transactions of the American Mathematical Society

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The FBI transform on compact ${\mathcal{C}^\infty}$ manifolds

Authors: Jared Wunsch and Maciej Zworski
Journal: Trans. Amer. Math. Soc. 353 (2001), 1151-1167
MSC (2000): Primary 35A22; Secondary 58J40, 81R30
Published electronically: November 8, 2000
MathSciNet review: 1804416
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We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact ${\mathcal{C}^\infty}$ manifold $M$. The FBI transform is a generalization of the classical notion of the wave-packet transform. We discuss the mapping properties of the FBI transform and its relationship to the calculus of pseudodifferential operators on $M$. We also describe the microlocal properties of its range in terms of the ``scattering calculus'' of pseudodifferential operators on the noncompact manifold $T^* M$.

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  • 1. Bargmann, V. On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, (1961) 187-214. MR 28:486
  • 2. Boutet de Monvel, L. and Guillemin, V., The spectral theory of Toeplitz operators. Annals of Mathematics Studies, 99, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 85j:58141
  • 3. Boutet de Monvel, L. and Sjöstrand, J., Sur la singularité des noyaux de Bergman et de Szegö, Journées: Équations aux Dérivées Partielles de Rennes (1975),123-164. Asterisque, 34-35 Soc. Math. France, Paris, 1976. MR 58:28684
  • 4. Córdoba, A. and Fefferman, C., Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3 (1978), 979-1005. MR 80a:35117
  • 5. Cordes, H. O., A global parametrix for pseudodifferential operators over ${\mathbb{R} }^n$ with applications, preprint No. 90, SFB 72, Bonn, 1976.
  • 6. Delort, J.-M., F.B.I. transformation. Second microlocalization and semilinear caustics. Lecture Notes in Mathematics 1522, Springer-Verlag, Berlin, 1992. MR 93i:35010
  • 7. Dimassi, M. and Sjöstrand, J., Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Note Ser., 268, Cambridge Univ. Press, Cambridge, 1999. CMP 2000:07
  • 8. Folland, G., Harmonic analysis in phase space, Ann. of Math. Studies, 122, Princton Univ. Press, Princeton, NJ, 1989. MR 92k:22017
  • 9. Golse, F., Leichtnam, E., and Stenzel, M. Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds. Ann. Sci. École Norm. Sup. 29 (1996), 669-736. MR 97h:58153
  • 10. Guillemin, V., Toeplitz operators in $ n$ dimensions. Integral Equations and Operator Theory, 7 (1984), 145-205. MR 86i:58130
  • 11. Helffer, B. and Sjöstrand, J. Resonances en limite semi-classique. Mém. Soc. Math. France (N.S.) 24-25 (1986).MR 88i:81025
  • 12. Hörmander, L., Linear partial differential equations, v.1, Springer Verlag, Berlin.
  • 13. Hörmander, L., Quadratic hyperbolic operators. Microlocal analysis and applications (Montecatini Terme, 1989), 118-160, Lecture Notes in Math. 1495, Springer, Berlin, 1991. MR 93k:35187
  • 14. Iagolnitzer, D., Microlocal essential support of a distribution and decomposition theorems - an introduction. in Hyperfunctions and theoretical physics (Rencontre, Nice, 1973), 121-132. Lecture Notes in Math., 449, Springer Verlag, Berlin, 1975. MR 52:11583
  • 15. Lebeau, G., Fonctions harmoniques et spectre singulier. Ann. Sci. École Norm. Sup. 13 (1980), 269-291. MR 81m:58072
  • 16. Martinez, A., Estimates on complex interactions in phase space. Math. Nachr. 167 (1994), 203-254. MR 95g:81025
  • 17. Melin, A. and Sjöstrand, J., ``Fourier integral operators with complex-valued phase functions,'' in Fourier integral operators and partial differential equations ed. J. Chazarain, Lecture Notes in Math. 459, Springer-Verlag, Berlin, 1975, 120-223. MR 55:4290
  • 18. Melrose, R. B., Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, Spectral and scattering theory (M. Ikawa, ed.), Marcel Dekker, 1994, 85-130. MR 95k:58168
  • 19. Melrose, R. B. and Zworski, M., Scattering metrics and geodesic flow at infinity. Invent. Math. 124 (1996), 389-436. MR 96k:58230
  • 20. Parenti, C., Operatori pseudodifferentiali in ${\mathbb{R} }^n$ e applicazioni, Ann. Math. Pura Appl. 93 (1972), 359-389. MR 55:10838
  • 21. Schrohe, E., Spaces of weighted symbols and weighted Sobolev spaces on manifolds, Pseudodifferential operators, Proceedings, Oberwolfach 1986, Lecture Notes in Mathematics 1256, Springer-Verlag, 1987, 360-377. MR 89g:58200
  • 22. Sjöstrand, J., Singularités analytiques microlocales. Astérisque, 95 (1982), 1-166. MR 84m:58151
  • 23. Sjöstrand, J., Lecture Notes, Lund University, 1985-86.
  • 24. Sjöstrand, J., Geometric bounds on the density of resonances for semi-classical problems. Duke Math. J., 60 (1990), 1-57. MR 91e:35166
  • 25. Sjöstrand, J., Density of resonances for strictly convex analytic obstacles. With an appendix by M. Zworski, Can. J. Math., 48 (1996), 397-447. MR 97j:35117
  • 26. Sjöstrand, J. and Zworski, M., The complex scaling method for scattering by strictly convex obstacles, Ark, Mat. 33 (1995), 135-172. MR 96f:35127
  • 27. Shubin, M. A., Pseudodifferential operators in ${\mathbb{R} }^n$, Dokl. Akad. Nauk SSSR, 196 (1971), 316-319, Soviet Math. Dokl. 12, No.1 (1971), 147-151. MR 42:8341
  • 28. Toth, J., Eigenfunction decay estimates in the quantum integrable case. Duke Math. J. 93 (1998), 231-255; 96 (1999), 469. MR 2000e:58041a,b
  • 29. Zworski, M., Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Invent. Math. 136 (1999), 353-409. CMP 99:12

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Additional Information

Jared Wunsch
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651

Maciej Zworski
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720

Keywords: FBI transform, Fourier-Bros-Iagolnitzer transformation, wave-packet
Received by editor(s): October 26, 1999
Published electronically: November 8, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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