Quantizations and symbolic calculus over the p-adic numbers
Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 997-1053.

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.

We develop the basic theory of the Weyl symbolic calculus of pseudodifferential operators over the p-adic numbers. We apply this theory to the study of elliptic operators over the p-adic numbers and determine their asymptotic spectral behavior.

@article{AIF_1993__43_4_997_0,
     author = {Haran, Shai},
     title = {Quantizations and symbolic calculus over the $p$-adic numbers},
     journal = {Annales de l'Institut Fourier},
     pages = {997--1053},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {4},
     year = {1993},
     doi = {10.5802/aif.1363},
     mrnumber = {95m:22004},
     zbl = {0974.22009},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1363/}
}
TY  - JOUR
AU  - Haran, Shai
TI  - Quantizations and symbolic calculus over the $p$-adic numbers
JO  - Annales de l'Institut Fourier
PY  - 1993
SP  - 997
EP  - 1053
VL  - 43
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.1363/
DO  - 10.5802/aif.1363
LA  - en
ID  - AIF_1993__43_4_997_0
ER  - 
%0 Journal Article
%A Haran, Shai
%T Quantizations and symbolic calculus over the $p$-adic numbers
%J Annales de l'Institut Fourier
%D 1993
%P 997-1053
%V 43
%N 4
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.1363/
%R 10.5802/aif.1363
%G en
%F AIF_1993__43_4_997_0
Haran, Shai. Quantizations and symbolic calculus over the $p$-adic numbers. Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 997-1053. doi : 10.5802/aif.1363. https://www.numdam.org/articles/10.5802/aif.1363/

[1] V. Bargmann, On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Comm. Pure Appl. Math., 14 (1961), 187-214. | MR | Zbl

[2] R. Beals, A General Calculus of Pseudodifferential Operators, Duke Math. J., 42 (1975), 1-42. | MR | Zbl

[3] J. Bergh, J. Lófström, Interpolation Spaces, Berlin-Heidelberg-New York, Springer, 1976. | Zbl

[4] F.A. Berezin, Wick and anti-Wick Operator Symbols, Math. USSR Sb., 15 (1971), 577-606. | Zbl

[5] A. Calderón, R. Vaillancourt, On the Boudedness of Pseudodifferential Operators, J. Math. Soc. Japan, 23 (1971), 374-378. | MR | Zbl

[6] P. Cartier, Quantum Mechanical Commutation Relations and Theta Functions, Proc. Symp. Pure Math., 9, AMS, Providence, 1966, 361-383. | MR | Zbl

[7] A. Córdoba, C. Fefferman, Wave Packets and Fourier Integral Operators, Comm. Partial Diff. Eq., 3 (1978), 979-1005. | MR | Zbl

[8] C. Fefferman, D.H. Phong, The Uncertainty Principle and Sharp Gårding Inequalities, Comm. Pure. Appl. Math., 34 (1981), 285-331. | MR | Zbl

[9] G.B. Folland, Harmonic Analysis in Phase Space, New Jersey, Princeton University Press, 1989. | MR | Zbl

[10] L. Gårding, On the Asymptotic of the Eigenvalues and Eigenfunctions of Elliptic Differential Operators, Math. Scand., 1 (1953), 237-255. | MR | Zbl

[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 | Zbl

[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. | Numdam | MR | Zbl

[13] V. Guillemin, S. Sternberg, The Metaplectic Representation, Weyl Operators, and Spectral Theory, J. Funct. Anal., 42 (1981), 128-225. | MR | Zbl

[14] S. Haran, Riesz Potentials and Explicit Sums in Arithmetic, Invent. Math., 101 (1990), 697-703. | MR | Zbl

[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). | Numdam | MR | Zbl

[17] B. Helffer, Théorie spectrale pour des opérateurs globalement elliptiques, Astérisque, 112 (1984). | Numdam | MR | Zbl

[18] L. Hörmander, The Weyl Calculus of Pseudodifferential Operators, Comm. Pure Appl. Math., 32 (1979), 359-443. | MR | Zbl

[19] L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer, Berlin-Heidelberg-New-York Tokyo, 1985. | Zbl

[20] L. Hörmander, On the Asymptotic Distribution of Eigenvalues of Pseudodifferential Operators in ℝn, Arkiv for Math., 17 (2) (1979), 296-313. | Zbl

[21] R. Howe, Quantum Mechanics and Partial Differential Equations, J. Funct. Anal., 38 (1980), 188-254. | MR | Zbl

[22] R. Howe, Theta Series and Invariant Theory, Proc. Symp. Pure Math. 33, AMS Providence 1979, part. 1, 275-285. | MR | Zbl

[23] R. Howe, On the Role of the Heisenberg Group in Harmonic Analysis, Bull. AMS, 3 (1980), 821-843. | MR | Zbl

[24] A.W. Knapp, E.M. Stein, Intertwining Operators for Semisimple Groups, Ann. of Math., 93 (1971), 489-578. | MR | Zbl

[25] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Series, 1976. | MR | Zbl

[26] J. Peetre, The Weyl Transform and Laguerre Polynomials, Le Mathematiche (Catania), 27 (1972), 301-323. | MR | Zbl

[27] D. Robert, Propriétés spectrales d'opérateurs pseudodifférentiels, Comm. Partial Diff. Eq., 3 (1978), 755-826. | MR | Zbl

[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

[31] M.H. Taibleson, Fourier Analysis on Local Fields, Princeton Univ. Press, 1975. | MR | Zbl

[32] M.E. Taylor, Noncommutative Harmonic Analysis, AMS Providence, 1986. | MR | Zbl

[33] F. Treves, Topological Vector Spaces, Distribution, and Kernels, Academic Press, New York, 1967. | MR | Zbl

[34] H. Triebel, Theory of Functions Spaces, Monogr. in Math. 78, Basel-Boston-Stuttgart, Birkhäuser, 1983. | MR | Zbl

[35] A. Voros, An Algebra of Pseudodifferential Operators and the Asymptotics of Quantum Mechanics, J. Funct. Anal., 29 (1978), 104-132. | MR | Zbl

[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 | Zbl

[37] H. Weyl, The Theory of Groups and Quantum Mechanics, New York, Dover, 1950. | Zbl

[38] R. Howe, The Oscillator Semigroup, Proc. Symp. Pure Math., 48 (1988), 61-132. | MR | Zbl

[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 | Zbl

[40] A. Unterberger, J. Unterberger, Quantification et analyse pseudodifférentielle, Ann. Scient. Écol. Norm. Sup., 21 (1988), 133-158. | Numdam | MR | Zbl

[41] A. Unterberger, J. Unterberger, Série principale et quantification, C.R. Acad. Sci. Paris, 312, Série 1 (1991), 729-734. | MR | Zbl

[42] V.S. Vladimirov, I.V. Volovich, p-adic Quantum Mechanics, Comm. Math. Phys., 123 (1989), 659-676. | MR | Zbl

  • Bastianoni, Federico; Cordero, Elena Quasi-Banach modulation spaces and localization operators on locally compact abelian groups, Banach Journal of Mathematical Analysis, Volume 16 (2022) no. 4 | DOI:10.1007/s43037-022-00205-6
  • Chacón-Cortés, L. F.; Garcia-Bibiano, C. A.; Zúñiga-Galindo, W. A. Local Well-Posedness of the Cauchy Problem for a p-Adic Nagumo-Type Equation, p-Adic Numbers, Ultrametric Analysis and Applications, Volume 14 (2022) no. 4, p. 279 | DOI:10.1134/s2070046622040021
  • Velasquez-Rodriguez, J. P. Hörmander Classes of Pseudo-Differential Operators over the Compact Group of p-Adic Integers, p-Adic Numbers, Ultrametric Analysis and Applications, Volume 12 (2020) no. 2, p. 134 | DOI:10.1134/s2070046620020053
  • Khrennikov, Andrei Yu.; Kochubei, Anatoly N. p-Adic Analogue of the Porous Medium Equation, Journal of Fourier Analysis and Applications, Volume 24 (2018) no. 5, p. 1401 | DOI:10.1007/s00041-017-9556-4
  • Vourdas, Apostolos A Quantum System with Positions in the Profinite Group Zp Z p, Finite and Profinite Quantum Systems (2017), p. 161 | DOI:10.1007/978-3-319-59495-8_11
  • Zúñiga-Galindo, W. A. p-Adic Analysis: Essential Ideas and Results, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 1 | DOI:10.1007/978-3-319-46738-2_1
  • Zúñiga-Galindo, W. A. Fundamental Solutions for Pseudodifferential Operators, and Equations of Schrödinger Type, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 127 | DOI:10.1007/978-3-319-46738-2_5
  • Zúñiga-Galindo, W. A. Parabolic-Type Equations and Markov Processes, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 13 | DOI:10.1007/978-3-319-46738-2_2
  • Zúñiga-Galindo, W. A. Pseudodifferential Equations of Klein-Gordon Type, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 145 | DOI:10.1007/978-3-319-46738-2_6
  • Zúñiga-Galindo, W. A. Final Remarks and Some Open Problems, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 167 | DOI:10.1007/978-3-319-46738-2_7
  • Zúñiga-Galindo, W. A. Non-Archimedean Parabolic-Type Equations with Variable Coefficients, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 43 | DOI:10.1007/978-3-319-46738-2_3
  • Zúñiga-Galindo, W. A. Parabolic-Type Equations and Markov Processes on Adeles, Pseudodifferential Equations Over Non-Archimedean Spaces, Volume 2174 (2016), p. 79 | DOI:10.1007/978-3-319-46738-2_4
  • Beltiţă, Ingrid; Beltiţă, Daniel; Măntoiu, Marius Symbol calculus of square-integrable operator-valued maps, Rocky Mountain Journal of Mathematics, Volume 46 (2016) no. 6 | DOI:10.1216/rmj-2016-46-6-1795
  • Gayral, Victor; Jondreville, David Deformation quantization for actions of Qpd, Journal of Functional Analysis, Volume 268 (2015) no. 11, p. 3357 | DOI:10.1016/j.jfa.2014.12.017
  • Măntoiu, M.; Purice, R. On Fréchet–Hilbert algebras, Archiv der Mathematik, Volume 103 (2014) no. 2, p. 157 | DOI:10.1007/s00013-014-0675-8
  • Beltiţă, Ingrid; Beltiţă, Daniel; Pascu, Mihai Boundedness for Pseudo-differential Calculus on Nilpotent Lie Groups, Geometric Methods in Physics (2013), p. 87 | DOI:10.1007/978-3-0348-0645-9_7
  • Vourdas, A Quantum mechanics on profinite groups and partial order, Journal of Physics A: Mathematical and Theoretical, Volume 46 (2013) no. 4, p. 043001 | DOI:10.1088/1751-8113/46/4/043001
  • Galeano-Peñaloza, J.; Zúñiga-Galindo, W.A. Pseudo-differential operators with semi-quasielliptic symbols over p-adic fields, Journal of Mathematical Analysis and Applications, Volume 386 (2012) no. 1, p. 32 | DOI:10.1016/j.jmaa.2011.07.040
  • Kaneko, Hiroshi Besov space and trace theorem on a local field and its application, Mathematische Nachrichten, Volume 285 (2012) no. 8-9, p. 981 | DOI:10.1002/mana.201000026
  • Vourdas, A. Quantum mechanics on Q/ZQ/Z, Journal of Mathematical Physics, Volume 52 (2011) no. 6 | DOI:10.1063/1.3597555
  • Zúñiga-Galindo, W. A. Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields, P-Adic Numbers, Ultrametric Analysis, and Applications, Volume 3 (2011) no. 4, p. 344 | DOI:10.1134/s207004661104008x
  • Kaneko, Hiroshi; Tsuzuki, Yoichi A lift of spatially inhomogeneous Markov process to extensions of the field ofp-adic numbers, Stochastic Processes and their Applications, Volume 121 (2011) no. 2, p. 394 | DOI:10.1016/j.spa.2010.10.008
  • Vourdas, A. Totally Disconnected and Locally Compact Heisenberg-Weyl Groups, Journal of Fourier Analysis and Applications, Volume 16 (2010) no. 5, p. 748 | DOI:10.1007/s00041-010-9125-6
  • Kaneko, H. Fractal theoretic aspects of local field, P-Adic Numbers, Ultrametric Analysis, and Applications, Volume 1 (2009) no. 1, p. 51 | DOI:10.1134/s2070046609010051
  • Gröchenig, Karlheinz; Strohmer, Thomas Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2007 (2007) no. 613 | DOI:10.1515/crelle.2007.094
  • Jorgenson, Jay; Lang, Serge The Ubiquitous Heat Kernel, Mathematics Unlimited — 2001 and Beyond (2001), p. 655 | DOI:10.1007/978-3-642-56478-9_34
  • Bibliography, Pseudo-Differential Equations And Stochastics Over Non-Archimedean Fields (2001) | DOI:10.1201/9780203908167.axa
  • Кочубей, Анатолий Наумович; Kochubei, Anatoly Naumovich Фундаментальные решения псевдодифференциальных уравнений, связанных с p-адическими квадратичными формами, Известия Российской академии наук. Серия математическая, Volume 62 (1998) no. 6, p. 103 | DOI:10.4213/im222

Cité par 28 documents. Sources : Crossref