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Shai Haran
Quantizations and symbolic calculus over the $p$-adic numbers
Annales de l'institut Fourier, 43 no. 4 (1993), p. 997-1053, doi: 10.5802/aif.1363
Article PDF | Analyses MR 95m:22004 | Zbl 0974.22009 | 1 citation dans Cedram

Résumé - Abstract

Nous développons la théorie du calcul symbolique des opérateurs pseudo-différentiels de Weyl sur les nombres $p$-adiques. Nous appliquons cette théorie à l’étude des opérateurs globalement elliptiques sur les nombres $p$-adiques et nous déterminons de façon exacte le comportement asymptotique de leur spectre.

Bibliographie

[1] V. BARGMANN, On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Comm. Pure Appl. Math., 14 (1961), 187-214.  MR 28 #486 |  Zbl 0107.09102
[2] R. BEALS, A General Calculus of Pseudodifferential Operators, Duke Math. J., 42 (1975), 1-42. Article |  MR 51 #3972 |  Zbl 0343.35078
[3] J. BERGH, J. LÓFSTRÖM, Interpolation Spaces, Berlin-Heidelberg-New York, Springer, 1976.  Zbl 0344.46071
[4] F.A. BEREZIN, Wick and anti-Wick Operator Symbols, Math. USSR Sb., 15 (1971), 577-606.  Zbl 0247.47018
[5] A. CALDERÓN, R. VAILLANCOURT, On the Boudedness of Pseudodifferential Operators, J. Math. Soc. Japan, 23 (1971), 374-378. Article |  MR 44 #2096 |  Zbl 0203.45903
[6] P. CARTIER, Quantum Mechanical Commutation Relations and Theta Functions, Proc. Symp. Pure Math., 9, AMS, Providence, 1966, 361-383.  MR 35 #7654 |  Zbl 0178.28401
[7] A. CÓRDOBA, C. FEFFERMAN, Wave Packets and Fourier Integral Operators, Comm. Partial Diff. Eq., 3 (1978), 979-1005.  MR 80a:35117 |  Zbl 0389.35046
[8] C. FEFFERMAN, D.H. PHONG, The Uncertainty Principle and Sharp Gårding Inequalities, Comm. Pure. Appl. Math., 34 (1981), 285-331.  MR 82j:35140 |  Zbl 0458.35099
[9] G.B. FOLLAND, Harmonic Analysis in Phase Space, New Jersey, Princeton University Press, 1989.  MR 92k:22017 |  Zbl 0682.43001
[10] L. GÅRDING, On the Asymptotic of the Eigenvalues and Eigenfunctions of Elliptic Differential Operators, Math. Scand., 1 (1953), 237-255.  MR 16,366b |  Zbl 0053.39102
[11] S. GELBART, Weil's Representation and the Spectrum of the Metaplectic Group, Lectures Notes in Math. Springer 530, Berlin-Heidelberg-New York, 1976.  MR 54 #12654 |  Zbl 0365.22017
[12] A. GROSSMAN, G. LOUPIAS, E.M. STEIN, An Algebra of Pseudodifferential Operators and Quantum Mechanics in Phase Space, Ann. Inst. Fourier (Grenoble), 18-2 (1968), 343-368. Cedram |  MR 42 #2327 |  Zbl 0176.45102
[13] V. GUILLEMIN, S. STERNBERG, The Metaplectic Representation, Weyl Operators, and Spectral Theory, J. Funct. Anal., 42 (1981), 128-225.  MR 83i:58101 |  Zbl 0469.58017
[14] S. HARAN, Riesz Potentials and Explicit Sums in Arithmetic, Invent. Math., 101 (1990), 697-703.  MR 91g:11132 |  Zbl 0788.11055
[15] S. HARAN, Index Theory, Potential Theory, and the Riemann Hypothesis Proc. Durham Symp. on L-functions and Arithmetic, Cambridge Univ. Press, 1991.
[16] S. HARAN, Analytic Potential Theory over the p-adics, Ann. Inst. Fourier (Grenoble), 43-4 (1993). Cedram |  MR 95c:11141 |  Zbl 0847.31006
[17] B. HELFFER, Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque, 112 (1984).  MR 86d:35151 |  Zbl 0541.35002
[18] L. HÖRMANDER, The Weyl Calculus of Pseudodifferential Operators, Comm. Pure Appl. Math., 32 (1979), 359-443.  MR 80j:47060 |  Zbl 0388.47032
[19] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators, III, Springer, Berlin-Heidelberg-New-York Tokyo, 1985.  Zbl 0601.35001
[20] L. HÖRMANDER, On the Asymptotic Distribution of Eigenvalues of Pseudodifferential Operators in ℝn, Arkiv for Math., 17 (2) (1979), 296-313.  Zbl 0436.35064
[21] R. HOWE, Quantum Mechanics and Partial Differential Equations, J. Funct. Anal., 38 (1980), 188-254.  MR 83b:35166 |  Zbl 0449.35002
[22] R. HOWE, Theta Series and Invariant Theory, Proc. Symp. Pure Math. 33, AMS Providence 1979, part. 1, 275-285.  MR 81f:22034 |  Zbl 0423.22016
[23] R. HOWE, On the Role of the Heisenberg Group in Harmonic Analysis, Bull. AMS, 3 (1980), 821-843. Article |  MR 81h:22010 |  Zbl 0442.43002
[24] A.W. KNAPP, E.M. STEIN, Intertwining Operators for Semisimple Groups, Ann. of Math., 93 (1971), 489-578.  MR 57 #536 |  Zbl 0257.22015
[25] J. PEETRE, New Thoughts on Besov Spaces, Duke Univ. Math. Series, 1976.  MR 57 #1108 |  Zbl 0356.46038
[26] J. PEETRE, The Weyl Transform and Laguerre Polynomials, Le Mathematiche (Catania), 27 (1972), 301-323.  MR 49 #5426 |  Zbl 0276.44005
[27] D. ROBERT, Propriétés spectrales d'opérateurs pseudodifférentiels, Comm. Partial Diff. Eq., 3 (1978), 755-826.  MR 80b:35112 |  Zbl 0392.35056
[28] R.T. SEELEY, The Complex Powers of an Elliptic Operator, Proc. Symp. Pure Math. 10, AMS, Providence, 1967, 308-315.
[29] J.-P. SERRE, Local Fields, Springer, Berlin-Heidelberg-New York, 1979.
[30] M.A. SUBIN, Pseudodifferential Operators and Spectral Theory, Nauka, Moscow, 1978.  Zbl 0451.47064
[31] M.H. TAIBLESON, Fourier Analysis on Local Fields, Princeton Univ. Press, 1975.  MR 58 #6943 |  Zbl 0319.42011
[32] M.E. TAYLOR, Noncommutative Harmonic Analysis, AMS Providence, 1986.  MR 88a:22021 |  Zbl 0604.43001
[33] F. TREVES, Topological Vector Spaces, Distribution, and Kernels, Academic Press, New York, 1967.  MR 37 #726 |  Zbl 0171.10402
[34] H. TRIEBEL, Theory of Functions Spaces, Monogr. in Math. 78, Basel-Boston-Stuttgart, Birkhäuser, 1983.  MR 86j:46026 |  Zbl 0546.46027
[35] A. VOROS, An Algebra of Pseudodifferential Operators and the Asymptotics of Quantum Mechanics, J. Funct. Anal., 29 (1978), 104-132.  MR 58 #14697 |  Zbl 0386.47031
[36] A. WEIL, Sur certains groupes d'opérateurs unitaires, Acta. Math., 111 (1964), 143-211; also in Weil's Œuvres Scientifiques, vol. III, 1-69, Springer, Berlin-Heidelberg-New York, 1980.  MR 29 #2324 |  Zbl 0203.03305
[37] H. WEYL, The Theory of Groups and Quantum Mechanics, New York, Dover, 1950.  Zbl 0041.56804
[38] R. HOWE, The Oscillator Semigroup, Proc. Symp. Pure Math., 48 (1988), 61-132.  MR 90f:22014 |  Zbl 0687.47034
[39] A. UNTERBERGER, J. UNTERBERGER, La serie discrète de SL(2,ℝ) et les opérateurs pseudo-différentiels sur une demi-droite, Ann. Scient. Écol. Norm. Sup., 17 (1984), 83-116. Numdam |  MR 86c:22026 |  Zbl 0549.35119
[40] A. UNTERBERGER, J. UNTERBERGER, Quantification et analyse pseudodifférentielle, Ann. Scient. Écol. Norm. Sup., 21 (1988), 133-158. Numdam |  MR 89h:58187 |  Zbl 0646.58025
[41] A. UNTERBERGER, J. UNTERBERGER, Série principale et quantification, C.R. Acad. Sci. Paris, 312, Série 1 (1991), 729-734.  MR 92c:22028 |  Zbl 0739.22014
[42] V.S. VLADIMIROV, I.V. VOLOVICH, p-adic Quantum Mechanics, Comm. Math. Phys., 123 (1989), 659-676. Article |  MR 90h:81049 |  Zbl 0688.22004
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