Archive for Rational Mechanics and Analysis

, Volume 226, Issue 3, pp 1161–1207 | Cite as

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

  • Michael RuzhanskyEmail author
  • Niyaz Tokmagambetov
Open Access


Given a Hilbert space \({\mathcal{H}}\), we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on \({\mathcal{H}}\). We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on \({\mathbb{R}^n}\), uniformly elliptic operators of different orders on domains, Hörmander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.


  1. 1.
    Abreu, L.D.: Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions. Appl. Comput. Harmon. Anal. 29, 287–302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abreu, L.D.; Balazs, P.; de Gosson, M.; Mouayn, Z.: Discrete coherent states for higher Landau levels. Ann. Phys. 363, 337–353 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ali, S.T.; Bagarello, F.; Gazeau, J.P.: Quantizations from reproducing kernel spaces. Ann. Phys. 332, 127–142 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bari, N.K.: Biorthogonal systems and bases in Hilbert space. Moskov. Gos. Univ. Učenye Zapiski Matematika 148(4), 69–107 (1951)MathSciNetGoogle Scholar
  5. 5.
    Bergeron, H.; Gazeau, J.P.: Integral quantizations with two basic examples. Ann. Phys. 344, 43–68 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonfiglioli, A.; Lanconelli, E.; Uguzzoni, F.: Stratified Lie groups and potential theory for their sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)zbMATHGoogle Scholar
  7. 7.
    Bronšten, M.D.: The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity. Trudy Moskov. Mat. Obshch. 41, 83–99 (1980)MathSciNetGoogle Scholar
  8. 8.
    Bruno, T.; Calzi, M.: Weighted sub-Laplacians on Métivier groups: essential self-adjointness and spectrum. Proc. Am. Math. Soc. 145, 3579–3594 (2017)CrossRefzbMATHGoogle Scholar
  9. 9.
    Cotfas, N., Gazeau, J.P., Grorska, K.: Complex and real Hermite polynomials and related quantizations. J. Phys. A: Math. Theor., 43(30): 305304, 1–14, 2010Google Scholar
  10. 10.
    Cicognani, M.; Colombini, F.: A well-posed Cauchyproblem for an evolution equation with coefficients of low regularity. J. Differ. Equ. 254(8), 3573–3595 (2013)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Colombini, F., De Giorgi, E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 6(3):511–559, 1979Google Scholar
  12. 12.
    Colombini, F.; Del Santo, D.; Reissig, M.: On the optimal regularity of coefficients in hyperbolic Cauchy problems. Bull. Sci. Math. 127(4), 328–347 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Colombini, F., Jannelli, E., Spagnolo, S.: Nonuniqueness in hyperbolic Cauchy problems. Ann. Math. (2), 126(3), 495–524, 1987Google Scholar
  14. 14.
    Colombini, F.; Kinoshita, T.: On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order. J. Differ. Equ. 186(2), 394–419 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Colombini, F.; Spagnolo, S.: An example of a weakly hyperbolic Cauchy problem not well posed in \(C^{\infty }\). Acta Math. 148, 243–253 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D'Ancona, P., Spagnolo, S.: Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(1), 169–185, 1998Google Scholar
  17. 17.
    Dasgupta, A.; Ruzhansky, M.: Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces. Bull. Sci. Math. 138(6), 756–782 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dasgupta, A.; Ruzhansky, M.: Eigenfunction expansions of ultradifferentiable functions and ultradistributions. Trans. Am. Math. Soc. 368(12), 8481–8498 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Delgado, J.; Ruzhansky, M.; Tokmagambetov, N.: Schatten classes, nuclearity and nonharmonic analysis on compact manifolds with boundary. J. Math. Pures Appl. 107(6), 758–783 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fischer, V., Ruzhansky, M., Taranto, C.: On sub-Laplacian gevrey spaces. preprint.Google Scholar
  21. 21.
    Fock, V.: Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld. Z. Phys. A 47(5–6), 446–448 (1928)CrossRefzbMATHGoogle Scholar
  22. 22.
    Garetto, C.; Ruzhansky, M.: On the well-posedness of weakly hyperbolic equations with time-dependent coefficients. J. Differ. Equ. 253(5), 1317–1340 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Garetto, C.; Ruzhansky, M.: Weakly hyperbolic equations with non-analytic coefficients and lower order terms. Math. Ann. 357(2), 401–440 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Garetto, C.; Ruzhansky, M.: Wave equation for sums of squares on compact Lie groups. J. Differ. Equ. 258, 4324–4347 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Garetto, C.; Ruzhansky, M.: Hyperbolic second order equations with non-regular time dependent coefficients. Arch. Rational Mech. Anal. 217(1), 113–154 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gelfand, I.M.: Some questions of analysis and differential equations. Am. Math. Soc. Transl. (2)26, 201–219 (1963)MathSciNetGoogle Scholar
  27. 27.
    de Gosson, M.: Spectral properties of a class of generalized Landau operators. Comm. Partial Differ. Equ. 33(11), 2096–2104 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Helffer, B.; Robert, D.: Asymptotique des niveaux d'énergie pour des hamiltoniens à un degr é de liberté. Duke Math. J. 49(4), 853–868 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hörmann, G.; de Hoop, M.V.: Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients. Acta Appl. Math. 67(2), 173–224 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hörmann, G.; de Hoop, M.V.: Detection of wave front set perturbations via correlation: foundation for wave-equation tomography. Appl. Anal. 81(6), 1443–1465 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hurd, A.E.; Sattinger, D.H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Am. Math. Soc. 132, 159–174 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Haimi, A.; Hedenmalm, H.: The polyanalytic Ginibre ensembles. J. Stat. Phys. 153(1), 10–47 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Inglis, J.: Spectral inequalities for operators on H-type groups. J. Spectr. Theory 2, 79–105 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ismail, M.: Analytic properties of complex Hermite polynomials. Trans. Am. Math. Soc. 368(2), 1189–1210 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kawamoto, M.: Exponential decay property for eigenfunctions of Landau–Stark Hamiltonia. Rep. Math. Phys. 77(1), 129–140 (2016)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Kinoshita, T.; Spagnolo, S.: Hyperbolic equations with non-analytic coefficients. Math. Ann. 336, 551–569 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Korotyaev, E.; Pushnitski, A.: A trace formula and high–energy spectral asymptotics for the perturbed Landau Hamiltonian. J. Funct. Anal. 217, 221–248 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kinoshita, T.; Spagnolo, S.: Hyperbolic equations with non-analytic coefficients. Math. Ann. 336(3), 551–569 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Landau, L.: Diamagnetismus der Metalle. Z. Phys. A 64(9–10), 629–637 (1930)CrossRefzbMATHGoogle Scholar
  40. 40.
    Lungenstrass, T.; Raikov, G.: A trace formula for long-range perturbations of the Landau Hamiltonian. Ann. Henri Poincare 15, 1523–1548 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Matsumoto, H.: Classical and non-classical eigenvalue asymptotics for magnetic Schrödinger operators. J. Funct. Anal. 95, 460–482 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Namark, M.A.: Linear differential operators. Part II: Linear differential operators in Hilbert space. With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt. Frederick Ungar Publishing Co., New York, 1968Google Scholar
  43. 43.
    Nicola, F.; Rodino, L.: Global pseudo-differential calculus on Euclidean spaces, vol. 4. Pseudo-Differ. Oper. Theory Appl. Birkhäuser Verlag, Basel (2010)Google Scholar
  44. 44.
    Nakamura, Sh: Gaussian decay estimates for the eigenfunctions of magnetic Schrödinger operators. Commun. Partial Differ. Equ. 21(5-6), 993–1006 (1996)CrossRefzbMATHGoogle Scholar
  45. 45.
    Oberguggenberger, M.: Multiplication of distributions and applications to partial differential equations, volume 259 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, 1992Google Scholar
  46. 46.
    Perelomov, A.: Generalized Coherent States and Their Applications. Texts and Monographs in Physics. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  47. 47.
    Persson, M.: Eigenvalue asymptotics of the even-dimensional exterior Landau–Neumann Hamiltonian. Adv. Math. Phys., Article ID 873704, 2009Google Scholar
  48. 48.
    Pushnitski, A.; Raikov, G.; Villegas-Blas, C.: Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian. Comm. Math. Phys. 320, 425–453 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pushnitski, A.; Rozenblum, G.: Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain. Doc. Math. 12, 569–586 (2007)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Rozenblum, G.; Tashchiyan, G.: On the spectral properties of the perturbed Landau Hamiltonian. Comm. Partial Differ. Equ. 33, 1048–1081 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Ruzhansky, M., Turunen, V.: Pseudo-differential operators and symmetries. Background analysis and advanced topics, volume 2 of Pseudo-Differential Operators. Theory and Applications. Birkhäuser Verlag, Basel, 2010Google Scholar
  52. 52.
    Ruzhansky, M.; Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups, \(\rm SU(2)\), 3-sphere, and homogeneous spaces. Int. Math. Res. Not. IMRN 11, 2439–2496 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Ruzhansky, M.; Tokmagambetov, N.: Nonharmonic analysis of boundary value problems. Int. Math. Res. Not. IMRN 12, 3548–3615 (2016)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Ruzhansky, M.; Tokmagambetov, N.: Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field. Lett. Math. Phys. 107, 591–618 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Ruzhansky, M., Tokmagambetov, N.: Nonharmonic analysis of boundary value problems without WZ condition. Math. Model. Nat. Phenom. 12, 115–140, 2017. (arXiv:1610.02159)
  56. 56.
    Ruzhansky, M., Taranto, C.: Time-dependent wave equations on graded groups. arXiv:1705.03047
  57. 57.
    Sambou, D.: Lieb-Thirring type inequalities for non–self–adjoint perturbations of magnetic Schrödinger operators. J. Funct. Anal. 266, 5016–5044 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Schwartz, L.: Sur l'impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Shkalikov, A.A.: Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 120(6), 12–21 (1982)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2.–Farabi Kazakh National UniversityAlmatyKazakhstan

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