Advertisement

A Complete Spectral Analysis of Generalized Gribov–Intissar’s Operator in Bargmann Space

  • Abdelkader IntissarEmail author
  • Jean-Karim Intissar
Article
  • 4 Downloads

Abstract

In the Bargmann representation, we study mathematically rigorously some interesting spectral properties of generalized Gribov–Intissar’s operator: \(\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda } = \lambda ' a^{*^{p+1}} a^{p+1}} {+ \mu a^{*^{p}}a^{p} + i\lambda a^{*^{p}}(a + a^{*})a^{p}\, (p = 0, 1, 2\ldots )}\) where \(a^{*}\) and a are the creation and annihilation operators; \([a, a^{*} ] = \mathbb {I }\). \((\lambda ', \mu , \lambda ) \in \mathbb {R}^{3}\) are respectively the four coupling, the intercept and the triple coupling of Pomeron and \(i^{2} = - 1\). Firstly, the domain of the operator is defined precisely and it is shown that the minimal and the maximal version of the operator are identical; here as well as in many subsequent stages of the analysis we use extensively the representation of the operators in the Bargmann space of analytic functions. Then it is proved that \(\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}\) has compact resolvent and thus its spectrum consists of complex eigenvalues. Furthermore, \(\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}\) generates a strongly continuous semigroup such that for all \(t \ge 0\) and a constant \(c > 0\) the estimate \(\displaystyle {\mid \mid e^{-t\mathbb {H}_{\lambda ', \mu , \lambda }}\mid \mid \le e^{ct}}\) holds, and the operator \(\displaystyle { e^{-t(\mathbb {H}_{\lambda ', \mu , \lambda } + c)}}\) is compact for all \(t > 0\). Moreover, the solutions of the Cauchy problem \(\displaystyle {\frac{du}{dt} + \mathbb {H}_{\lambda ', \mu , \lambda }u = 0}\) can be expanded as a series in the eigenvectors of \(\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}\). Similar results concerning the associated semigroups are established for the operator \(\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}\) for \(\lambda ' = 0\). If \(p = 0\), \(\lambda ' = 0\) and \(\lambda \in i\mathbb {R}\) then \(\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}\) is the displaced harmonic oscillator. Secondly If \(p = 1\), the Reggeon field theory (Boreskov et al. in Phys Atomic Nucl 69(10):1765–1780, 2006; Gribov in Sov. Phys. JETP 26(2):414–423, 1968) is governed by \(\displaystyle { \mathbb {H}_{\lambda ', \mu ,\lambda }}\). In this case for \(\lambda '> 0 ,\mu > 0\), let \(\sigma (\lambda ',\mu , \lambda ) \ne 0\) be the smallest eigenvalue of \(H_{\lambda ',\mu ,\lambda }\), we show in this paper that \(\sigma (\lambda ',\mu , \lambda )\) is positive, increasing and analytic function on the whole real line with respect to \(\mu \) and that the spectral radius of \(H_{\lambda ',\mu ,\lambda }^{-1}\) converges to that of \(H_{0,\mu ,\lambda }^{-1}\) as \(\lambda '\) goes to zero. Hence we can exploit the structure of \(H_{\lambda ',\mu ,\lambda }^{-1}\) as \(\lambda '\) goes to zero to deduce the main results of Ando–Zerner established (Ando and Zerner in Commun Math Phys 93:123–139, 1984) on the function \(\sigma (0,\mu , \lambda )\). Thirdly, If \( \lambda ' = \mu = 0\), we consider the generalized operator \(\displaystyle {H^{p,m} = a^{*^{p}} (a^{m} + a^{*^{m}})a^{p}}\); \((p, m=1, 2,\ldots )\) of \(\displaystyle {-\frac{i}{\lambda }\mathbb {H}_{0, 0, \lambda }}\) acting on Bargmann space. For this operator, we find some conditions on the parameters p and m for that \(\displaystyle { H^{p,m}}\) to be completely indeterminate. It follows from these conditions that \(\displaystyle { H^{p,m}}\) is entire of the type minimal. And by applying the main result of the authors (Intissar and Intissar in Complex Anal Oper Theory 11(3):491–505, 2017), we show that \(\displaystyle { H^{p,m}}\) and \(\displaystyle { H^{p,m}+ H^{*^{p,m}}}\) are connected at the chaotic operators (where \(H^{*^{p,m}}\) is the adjoint of the \(H^{p,m}\)).

Keywords

Spectral theory Gribov–Intissar’s operators Semigroups Unbounded chaotic operators Entire operators Bargmann space Reggeon field theory 

Notes

References

  1. 1.
    Aimar, M.-T., Intissar, A., Paoli, J.-M.: Densité des vecteurs propres généralisés d’une classe d’opérateurs compacts non auto-adjoints et applications. Commun. Math. Phys. 156, 169–177 (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aimar, M.-T., Intissar, A., Paoli, J.-M.: Quelques nouvelles propriétés de régularité de l’opérateur de Gribov. Commun. Math. Phys. 172, 461–466 (1995)CrossRefzbMATHGoogle Scholar
  3. 3.
    Aimar, M.-T., Intissar, A., Paoli, J.-M.: Critères de Complétude des Vecteurs Propres Généralisés d’une Classe d’Opérateurs Non Auto-adjoints Compacts ou à Résolvante Compacte et Applications. Publ. Res. Inst. Math. Sci. Kyoto Univ. 32, 191–205 (1996)CrossRefzbMATHGoogle Scholar
  4. 4.
    Aimar, M.-T., Intissar, A., Intissar, J.-K.: On regularized trace formula of Gribov semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann representation. Complex Anal. Oper. Theory (2017).  https://doi.org/10.1007/s11785-017-0707-z zbMATHGoogle Scholar
  5. 5.
    Ando, T., Zerner, M.: Sur une valeur propre d’un opérateur. Commun. Math. Phys. 93, 123–139 (1984). (in French)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bargmann, V.: On Hilbert space of analytic functions and associated integral transform, part I. Commun. Pure Appl. Math. 14, 187–214 (1961)CrossRefzbMATHGoogle Scholar
  7. 7.
    Boreskov, K.G., Kaidalov, A.B., Kancheli, O.V.: Strong interactions at high energies in the Reggeon approch. Phys. Atomic Nucl. 69(10), 1765–1780 (2006)CrossRefGoogle Scholar
  8. 8.
    Bès, J., Chan, K., Seubert, S.S.: Chaotic unbounded differentiation operators. Integral Equ. Oper. Theory 40, 257–267 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chistyakov, A.L.: Deficiency numbers of symmetric operators in a direct sum of Hilbert spaces, I. Vestn. Mosk. Univ. Ser. Mat. Mekh. 3, 5–21 (1969)Google Scholar
  10. 10.
    Chistyakov, A.L.: Deficiency numbers of symmetric operators in a direct sum of Hilbert spaces, II. Vestn. Mosk. Univ. Ser. Mat. Mekh. 4, 3–5 (1969)Google Scholar
  11. 11.
    Chistyakov, A.L.: Deficiency indices of \(J_{m}\)-matrices and differential operators with polynomial coefficients. Mat. Sb. 4, 474–503 (1971)Google Scholar
  12. 12.
    Decarreau, A., Emamirad, H., Intissar, A.: Chaoticité de l’opérateur de Gribov dans l’espace de Bargmann. C. R. Acad. Sci. Paris 331 (Série I), 751–756 (2000)Google Scholar
  13. 13.
    Devinatz, A.: The deficciency index problem for ordinary selfadjoint differential operators. Bull. Am. Math. Soc. 79(6), 1109–1127 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Friedrichs, K.O.: Spectral Theory of Operators in Hilbert Space. Springer, Berlin (1973)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gelfond, A.O., Leontiev, A.F.: On a generalization of the Fourier series. Mat. Sbornik. 29(71), 477–500 (1951) (in Russian)Google Scholar
  16. 16.
    Gorbachuk, M.L., Gorbachuk, V.I.: Krein’s Lectures on Entire Operators. Operator Theory: Advances and Applications, vol. 97. Birkhaüser Verlag, Basel (1997)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gribov, V.: A reggeon diagram technique. Sov. Phys. JETP 26(2), 414–423 (1968)Google Scholar
  18. 18.
    Intissar, A., Le Bellac, M., Zerner, M.: Properties of the Hamiltonian of Reggeon field theory. Phys. Lett. B 113, 487–489 (1982)CrossRefGoogle Scholar
  19. 19.
    Intissar, A.: Etude spectrale d’une famille d’opérateurs non symétriques intervenant dans la théorie des champs des reggeons. Commun. Math. Phys. 113(2), 263–297 (1987)CrossRefzbMATHGoogle Scholar
  20. 20.
    Intissar, A.: Analyse de scattering d’un opérateur cubique de Heun dans l’espace de Bargmann. Commun. Math. Phys. 199, 243–256 (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Intissar, A.: Approximation of the semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann space. J. Math. Anal. Appl. 305, 669–689 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Intissar, A.: An elementary construction on nonlinear coherent states associated to generalized Bargmann spaces. Int. J. Math. Math. Sci. 2010, 15 (2010).  https://doi.org/10.1155/2010/357050 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Intissar, A.: On a chaotic weighted shift \(z^{p}\frac{d^{p+1}}{dz^{p+1}}\) of order \(p\) in Bargmann space. Adv. Math. Phys. (2011).  https://doi.org/10.1155/2011/471314 zbMATHGoogle Scholar
  24. 24.
    Intissar, A.: On a chaotic weighted shift \(z^{p}\mathbb{D}^{p+1}\) of order \(p\) in generalized Fock–Bargmann spaces. Math. Aeterna 3(7), 519–534 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Intissar, A.: A short note on the chaoticity of a weight shift on concrete orthonormal basis associated to some Fock–Bargmann space. J. Math. Phys. 55, 011502 (2014).  https://doi.org/10.1063/1.4861931 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Intissar, A.: Spectral analysis of non-selfadjoint Jacobi–Gribov operator and asymptotic analysis of its generalized eigenvectors. Adv. Math. (China) 44(3), 335–353 (2015).  https://doi.org/10.11845/sxjz.2013117b zbMATHGoogle Scholar
  27. 27.
    Intissar, A.: Regularized trace formula of magic Gribov operator on Bargmann space. J. Math. Anal. Appl. 437, 59–70 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Intissar, A.: On spectral approximation of unbounded Gribov–Intissar operators in Bargmann space. Adv. Math. (China) 46(1), 13–33 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Intissar, A., Intissar, J.K.: On chaoticity of the sum of chaotic shifts with their adjoints in Hilbert space and applications to some weighted shifts acting on some Fock–Bargmann spaces. Complex Anal. Oper. Theory 11(3), 491–505 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Intissar, A.: Analyse Fonctionnelle et Théorie Spectrale pour les Opérateurs Compacts Non Auto-Adjoints et exercices avec solutions, Editions CEPADUES (1997)Google Scholar
  31. 31.
    Intissar, A., Intissar, J.K.: Calcul différentiel, Fondement et applications, cours et exercices avec solutions, Editions CEPADUES (2017)Google Scholar
  32. 32.
    Jentzsch, P.: Uber Integralgleichungen mit positivem Kern. J. Reine Angw. Moth. 141, 235–244 (1912)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)CrossRefzbMATHGoogle Scholar
  34. 34.
    Kostyuchenko, A.G., Mirsoev, K.A.: Three-term recurrence relations with matrix coefficients, the completely indeterminate case. Mat. Zametki 63(5), 709–716 (1998)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kostyuchenko, A.G., Mirsoev, K.A.: Generalized Jacobi matrices and deficiency numbers of ordinary differential operators with polynomial coefficients. Funct. Anal. Appl. 33(1), 2537 (1999)CrossRefGoogle Scholar
  36. 36.
    Krein, M.G.: Infinite J-matrices and the matrix moment problem. Dokl. Akad. Nauk SSSR 69(3), 125–128 (1949)MathSciNetGoogle Scholar
  37. 37.
    Pazy, A.: Semi-Groups of Linear Operators and Applications to Partial Diffrential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  38. 38.
    Reed, M., Simon, B.: Analysis of Operators (Methods of Modern Mathematical Physics IV). Academic Press, New York (1978)zbMATHGoogle Scholar
  39. 39.
    Swanson, M.S.: Transition elements for a non-Hermitian quadratic Hamiltonian. J. Math. Phys. 45, 585601 (2004)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Schafer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefGoogle Scholar
  41. 41.
    Silva, L.O., Toloza, J.H.: The class of n-entire operators. J. Phys. A 46, 025202 (2013). (23 pp)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zerner, M.: Quelques propriétés spectrales des opérateurs positifs. J. Funct. Anal. 72, 381–417 (1987). (in French)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Equipe d’Analyse spectrale, UMR-CNRS no: 6134Université de Corse, Quartier GrossettiCortéFrance
  2. 2.Imperial College London, South Kensington Campus LondonLondonUK
  3. 3.Ecole Centrale ParisUniversité Paris-SaclayGif-sur-YvetteFrance
  4. 4.Le PradorMarseilleFrance

Personalised recommendations