Abstract
In this paper, we study a few spectral properties of a non-symmetrical operator arising in the Gribov theory.
The first and second section are devoted to Bargmann's representation and the study of general spectral properties of the operator:
whereA* j andA j ,j∈[1,N] are the creation and annihilation operators. In the third section, we restrict our study to the case of nul transverse dimension (N=1). Following the study done in [1], we consider the operator:
whereA* andA are the creation and annihilation operators.
For λ′>0 and λ′2≦μλ′+λ2. We prove that the solutions of the equationu′(t)+H λ′, μ,λ u(t)=0 are expandable in series of the eigenvectors ofH λ′,μ,λ fort>0.
In the last section, we show that the smallest eigenvalue σ(α) of the operatorH λ′,μ,λ,α is analytic in α, and thus admits an expansion: σ(α)=σ0+ασ1+α2σ2+..., where σ0 is the smallest eigenvalue of the operatorH λ′,μ,λ,0.
Similar content being viewed by others
Bibliographie
Ando, T., Zerner, M.: Sur une valeur propre d'un opérateur. Commun. Math. Physics93, 123–139 (1984)
Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. I. Commun. Pure App. Math.14, 187–214 (1961)
Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. II. Commun. Pure App. Math.20, 220–242 (1967)
Besov, LL'in, Nucol'skii: Integral representation of functions and imbedding theorem. V.I., Winston Willy 1978
Dieudonné, J.: Fondements de l'analyse moderne Fasicule XXVIII. Paris: Gauthier-Villars
Friedrichs, K.O.: Spectral theory of operators in Hilbert space. Berlin, Heidelberg, New York: Springer 1973
Friedrichs, K.O.: On the perturbation of continuous spectra, Comm. Appl. Math.1, 361–406 (1948)
Gohberg-Krein, -.: Introduction to the theory of linear non-self-adjoint operators.18, A.M.S. (1969)
Gribov, V.: J.E.T.P. (Sov. Phys.)26, 414 (1968)
Herbst, I.W.: Contraction semi-groups and the spectrum ofA 1⊗I+I⊗A 2. J. Operator Theory7, 64–78 (1982)
Intissar, A.: Etude spectrale d'une famille d'opérateurs non-symétriques intervenant dans le théorie des champs de Reggeons; Thèse D'Etat, Université de Nice (1986)
Intissar, A.: Sur une propriété spectrale d'un opérateur non symétrique intervenant dans la théorie de Regge, C.R. Acad. Sci. Paris T294, 715–718 (1982)
Intissar, A., Le Bellac, M., Zerner, M.: Properties of the Hamiltonian of reggeon field theory. Phys. Lett.113B, 487–489 (1982)
Intissar, A.: Diagonalisation d'opérateurs non aut-adjoints intervenant dans la théorie des champs des reggeons de Gribov. C.R. Acad. Sci. Paris T:304, No 2 Sér I, 43–46 (1987)
Intissar, A.: Quelques propriétés spectrales de l'hamiltonien de la théorie des champs de reggeons, C.R. Acad. Sci. Paris T304, No 3 Sér I, 63–66 (1987)
Intissar, A.: Sur une méthode de perturbation C.R. Acad. Sci. Paris T304, No 4 Série I, 95–98 (1987)
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
Kato, T.: Perturbation theory of semi-bounded operators. J. Math. Annal.0, 435–447 (1953)
Lidskii, V.B.: Summability of series in the principal vectors of non self-adjoint operators. Am. Math. Soc. Trans. Ser. 2,40, 193–228
Sz-Nagy, B.: Perturbations des transformations auto-adjointes dans l'espace de Hilbert. Comment. Math. Helv.19, 347–366 (1947)
Sz-Nagy, B.: Perturbations des transformations linéaires fermées. Acta. Sci. Math.14, 125–137 (1951)
Okazawa: Singular perturbations ofm-accretive operators; J. Math. Soc. Jpn.32, 19–44 (1980)
Pazy, A.: Semi-groups of linear operators and applications to partial diffrential equations. Berlin, Heidelberg, New York: Springer 1983
Rellich, F.: Perturbation theory of eigenvalue problems. Notes on Mathematics and its Applications. pp. 71–80. New York: Gordon and Breach 1969
Valiron, G.: Equations fonctionnellles — applications, p. 203. Paris: Masson 1950
Weidman, J.: Linear operators in Hilbert space. Berlin, Heidelberg, New York: Springer 1980
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Intissar, A. Etude spectrale d'une famille d'opérateurs non-symétriques intervenant dans la théorie des champs de reggeons. Commun.Math. Phys. 113, 263–297 (1987). https://doi.org/10.1007/BF01223514
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01223514