Quantizations and symbolic calculus over the $p$-adic numbers

Haran, Shai

Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 997-1053

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.

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.

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

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

[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. | Zbl | MR

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

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

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

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

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

[11] S. Gelbart, Weil's Representation and the Spectrum of the Metaplectic Group, Lectures Notes in Math. Springer 530, Berlin-Heidelberg-New York, 1976. | Zbl | MR

[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. | Zbl | MR | Numdam

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

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

[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). | Zbl | MR | Numdam

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

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

[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. | Zbl | MR

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

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

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

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

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

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

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

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

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

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