Annals of Global Analysis and Geometry

, Volume 36, Issue 3, pp 293–322 | Cite as

Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups

  • Ingrid Beltiţă
  • Daniel BeltiţăEmail author
Original Paper


We develop a pseudo-differential Weyl calculus on nilpotent Lie groups, which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose, we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie group and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.


Weyl calculus Magnetic field Lie group Semidirect product 

Mathematics Subject Classification (2000)

Primary 47G30 Secondary 22E25 22E65 35S05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson R.F.V.: The Weyl functional calculus. J. Funct. Anal. 4, 240–267 (1969)zbMATHCrossRefGoogle Scholar
  2. 2.
    Anderson R.F.V.: The multiplicative Weyl functional calculus. J. Funct. Anal. 9, 423–440 (1972)zbMATHCrossRefGoogle Scholar
  3. 3.
    Baguis P.: Semidirect products and the Pukanszky condition. J. Geom. Phys. 25(3–4), 245–270 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Beltiţă D.: Smooth Homogeneous Structures in Operator Theory. Monographs and Surveys in Pure and Applied Mathematics, Vol. 137. Chapman & Hall/CRC, Boca Raton, FL (2006)Google Scholar
  5. 5.
    Beltiţă I.: Inverse scattering in a layered medium. Comm. Partial Differential Equations 26(9–10), 1739–1786 (2001)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Beltiţă, I.: On an abstract radiation condition. In: Spectral and Scattering Theory and Related Topics (Kyoto, 2000) (pp 80–90). Sūrikaisekikenkyūsho Kōkyūroku No. 1208 (2001)Google Scholar
  7. 7.
    Boyarchenko M., Levendorski S.: Beyond the classical Weyl and Colin de Verdière’s formulas for Schrödinger operators with polynomial magnetic and electric fields. Ann. Inst. Fourier (Grenoble) 56(6), 1827–1901 (2006)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cahen B.: Quantification d’une orbite massive d’un groupe de Poincaré généralisé. C. R. Acad. Sci. Paris Sér. I Math. 325(7), 803–806 (1997)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Cahen B.: Quantification d’orbites coadjointes et théorie des contractions. J. Lie Theory 11(2), 257–272 (2001)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Cahen B.: Weyl quantization for semidirect products. Differential Geom. Appl. 25(2), 177–190 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Berkeley Mathematics Lecture Notes, vol. 10. American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, (1999)Google Scholar
  12. 12.
    Głowacki P.: A symbolic calculus and L 2-boundedness on nilpotent Lie groups. J. Funct. Anal. 206(1), 233–251 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Głowacki P.: The Melin calculus for general homogeneous groups. Ark. Mat. 45(1), 31–48 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Helffer B., Nourrigat J.: Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs. Progress in Mathematics, 58. Birkhäuser Boston Inc., Boston, MA (1985)Google Scholar
  15. 15.
    Hofmann K.H., Morris S.A.: Sophus Lie’s third fundamental theorem and the adjoint functor theorem. J. Group Theory 8(1), 115–133 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hofmann K.H., Morris S.A.: The Lie Theory of Connected Pro-Lie Groups. A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups. EMS Tracts in Mathematics, vol. 2. European Mathematical Society (EMS), Zürich (2007)Google Scholar
  17. 17.
    Hörmander L.: The Weyl calculus of pseudodifferential operators. Comm. Pure Appl. Math. 32(3), 360–444 (1979)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Hörmander L.: The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators. Reprint of the 1994 edition. Classics in Mathematics. Springer, Berlin (2007) Reprint of the 1994 editionGoogle Scholar
  19. 19.
    Howe R.E.: On a connection between nilpotent groups and oscillatory integrals associated to singularities. Pacific J. Math. 73(2), 329–363 (1977)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Iftimie V., Măntoiu M., Purice R.: Magnetic pseudodifferential operators. Publ. Res. Inst. Math. Sci. 43(3), 585–623 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Karasev M.V., Osborn T.A.: Quantum magnetic algebra and magnetic curvature. J. Phys. A 37(6), 2345–2363 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kirillov, A.A.: Unitary representations of nilpotent Lie groups. (Russian) Uspehi Mat. Nauk 17(4, 106), 57–110 (1962)Google Scholar
  23. 23.
    Kirillov A.A.: Elements of the Theory of Representations. Grundlehren der Mathematischen Wissenschaften, vol. 220. Springer-Verlag, Berlin (1976)Google Scholar
  24. 24.
    Kriegl A., Michor P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence, RI (1997)Google Scholar
  25. 25.
    Lang S.: Fundamentals of Differential Geometry (corrected second printing), Graduate Texts in Mathematics, vol. 191. Springer-Verlag, New York (2001)Google Scholar
  26. 26.
    Lichnerowicz A.: Characterization of Lie groups on the cotangent bundle of a Lie group. Lett. Math. Phys. 12(2), 111–121 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Manchon D.: Formule de Weyl pour les groupes de Lie nilpotents. J. Reine Angew. Math. 418, 77–129 (1991)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Măntoiu M., Purice R.: The magnetic Weyl calculus. J. Math. Phys. 45(4), 1394–1417 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Melin A.: Parametrix constructions for some classes of right-invariant differential operators on the Heisenberg group. Comm. Partial Differential Equations 6(12), 1363–1405 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Melin, A.: On the construction of fundamental solutions for differential operators on nilpotent groups. J. Équ. Dériv. Partielles, Saint-Jean-de-Monts 15 (1981)Google Scholar
  31. 31.
    Melin A.: Parametrix constructions for right invariant differential operators on nilpotent groups. Ann. Global Anal. Geom. 1(1), 79–130 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Miller K.G.: Invariant pseudodifferential operators on two-step nilpotent Lie groups. Michigan Math. J. 29(3), 315–328 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Miller K.G.: Invariant pseudodifferential operators on two-step nilpotent Lie groups. II. Michigan Math. J. 33(3), 395–401 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Moskowitz M., Sacksteder R.: The exponential map and differential equations on real Lie groups. J. Lie Theory 13(1), 291–306 (2003)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Neeb K.-H.: Towards a Lie theory of locally convex groups. Japan. J. Math. 1(2), 291–468 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Pedersen N.V.: Matrix coefficients and a Weyl correspondence for nilpotent Lie groups. Invent. Math. 118(1), 1–36 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Taylor M.E.: Noncommutative Harmonic Analysis. Mathematical Surveys and Monographs, vol. 22. American Mathematical Society, Providence, RI (1986)Google Scholar
  38. 38.
    Wildberger N.J.: Convexity and unitary representations of nilpotent Lie groups. Invent. Math. 98(2), 281–292 (1989)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

Personalised recommendations