×
Wunsch, Jared; Zworski, Maciej

The FBI transform on compact \({\mathcal{C}^\infty}\) manifolds. (English) Zbl 0974.35005

Trans. Am. Math. Soc. 353, No. 3, 1151-1167 (2001).
This paper can be considered as a comprehensive geometric treatment of Cordoba and Fefferman theory of wave-packet integral kernels; see A. Córdoba and C. Fefferman [Commun. Partial Differ. Equations 3, 979-1005 (1978; Zbl 0389.35046)]. In fact, a geometric theory of the FBI transform and a compact \(C^\infty\) manifold is presented, as the FBI transform is considered as a generalization of the classical notion of the wave-packet transform.
The authors point of view is similar to those of J. Sjöstrand [Can. J. Math. 48, No. 2, 397-447 (1996; Zbl 0863.35071)], but dropping analyticity assumption and constructing directly the orthogonal projection onto the range of the transform. An interconnection with Melrose’s “scattering calculus” of pseudodifferential operators on the noncompact manifold \(T^*M\) is described, too.
Reviewer: Stanco Dimiev (Sofia)

Cited in 10 Documents

MSC:

35A22 Transform methods (e.g., integral transforms) applied to PDEs
58J40 Pseudodifferential and Fourier integral operators on manifolds
81R30 Coherent states

Keywords:

wave packet; scattering calculus

Citations:

Zbl 0389.35046; Zbl 0863.35071
Cite Review PDF
Full Text: DOI

References:

[1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187 – 214. · Zbl 0107.09102 · doi:10.1002/cpa.3160140303
[2] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. · Zbl 0469.47021
[3] L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegő, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123 – 164. Astérisque, No. 34-35 (French). · Zbl 0344.32010
[4] Antonio Córdoba and Charles Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979 – 1005. · Zbl 0389.35046 · doi:10.1080/03605307808820083
[5] Cordes, H. O., A global parametrix for pseudodifferential operators over \({\mathbb{R} }^n\) with applications, preprint No. 90, SFB 72, Bonn, 1976.
[6] Jean-Marc Delort, F.B.I. transformation, Lecture Notes in Mathematics, vol. 1522, Springer-Verlag, Berlin, 1992. Second microlocalization and semilinear caustics. · Zbl 0760.35004
[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 · Zbl 0926.35002
[8] Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.43001
[9] Eric Leichtnam, François Golse, and Matthew Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 6, 669 – 736. · Zbl 0889.32037
[10] Victor Guillemin, Toeplitz operators in \? dimensions, Integral Equations Operator Theory 7 (1984), no. 2, 145 – 205. · Zbl 0561.47025 · doi:10.1007/BF01200373
[11] B. Helffer and J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) 24-25 (1986), iv+228 (French, with English summary). · Zbl 0631.35075
[12] Hörmander, L., Linear partial differential equations, v.1, Springer Verlag, Berlin. · Zbl 0191.10901
[13] Lars Hörmander, Quadratic hyperbolic operators, Microlocal analysis and applications (Montecatini Terme, 1989) Lecture Notes in Math., vol. 1495, Springer, Berlin, 1991, pp. 118 – 160. · Zbl 0761.35004 · doi:10.1007/BFb0085123
[14] D. Iagolnitzer, Appendix: Microlocal essential support of a distribution and decomposition theorems — an introduction, Hyperfunctions and theoretical physics (Rencontre, Nice, 1973; dédié à la mémoire de A. Martineau), Springer, Berlin, 1975, pp. 121 – 132. Lecture Notes in Math., Vol. 449.
[15] Gilles Lebeau, Fonctions harmoniques et spectre singulier, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 269 – 291 (French). · Zbl 0446.46035
[16] André Martinez, Estimates on complex interactions in phase space, Math. Nachr. 167 (1994), 203 – 254. · Zbl 0836.35135 · doi:10.1002/mana.19941670109
[17] Anders Melin and Johannes Sjöstrand, Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) Springer, Berlin, 1975, pp. 120 – 223. Lecture Notes in Math., Vol. 459. · Zbl 0306.42007
[18] Richard B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (Sanda, 1992) Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, New York, 1994, pp. 85 – 130. · Zbl 0837.35107
[19] Richard Melrose and Maciej Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1996), no. 1-3, 389 – 436. · Zbl 0855.58058 · doi:10.1007/s002220050058
[20] Cesare Parenti, Operatori pseudo-differenziali in \?\(^{n}\) e applicazioni, Ann. Mat. Pura Appl. (4) 93 (1972), 359 – 389. · Zbl 0291.35070 · doi:10.1007/BF02412028
[21] Elmar Schrohe, Spaces of weighted symbols and weighted Sobolev spaces on manifolds, Pseudodifferential operators (Oberwolfach, 1986) Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 360 – 377. · Zbl 0638.58026 · doi:10.1007/BFb0077751
[22] Johannes Sjöstrand, Singularités analytiques microlocales, Astérisque, 95, Astérisque, vol. 95, Soc. Math. France, Paris, 1982, pp. 1 – 166 (French).
[23] Sjöstrand, J., Lecture Notes, Lund University, 1985-86.
[24] Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1 – 57. · Zbl 0702.35188 · doi:10.1215/S0012-7094-90-06001-6
[25] Johannes Sjöstrand, Density of resonances for strictly convex analytic obstacles, Canad. J. Math. 48 (1996), no. 2, 397 – 447 (English, with English and French summaries). With an appendix by M. Zworski. · Zbl 0863.35072 · doi:10.4153/CJM-1996-022-9
[26] Johannes Sjöstrand and Maciej Zworski, The complex scaling method for scattering by strictly convex obstacles, Ark. Mat. 33 (1995), no. 1, 135 – 172. · Zbl 0839.35095 · doi:10.1007/BF02559608
[27] M. A. Šubin, Pseudodifferential operators in \?\(^{n}\), Dokl. Akad. Nauk SSSR 196 (1971), 316 – 319 (Russian).
[28] Toth, J., Eigenfunction decay estimates in the quantum integrable case. Duke Math. J. 93 (1998), 231-255; 96 (1999), 469. · Zbl 0941.58017
[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 · Zbl 1016.58014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.