A Complete Spectral Analysis of Generalized Gribov-Intissar's Operator in Bargmann Space

摘要

In the Bargmann representation, we study mathematically rigorously some interesting spectral properties of generalized Gribov-Intissar's operator: H,,=ap+1ap+1+apap+iap(a+a)ap(p=0,1,2...) where a and a are the creation and annihilation operators; [a,a]=I. (,,)R3 are respectively the four coupling, the intercept and the triple coupling of Pomeron and i2=-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 H,, has compact resolvent and thus its spectrum consists of complex eigenvalues. Furthermore, H,, generates a strongly continuous semigroup such that for all t0 and a constant c>0 the estimate ||e-tH,,||ect holds, and the operator e-t(H,,+c) is compact for all t>0. Moreover, the solutions of the Cauchy problem dudt+H,,u=0 can be expanded as a series in the eigenvectors of H,,. Similar results concerning the associated semigroups are established for the operator H,, for =0. If p=0, =0 and iR then H,, 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 H,,. In this case for >0,>0, let sigma(,,)0 be the smallest eigenvalue of H,,, we show in this paper that sigma(,,) is positive, increasing and analytic function on the whole real line with respect to and that the spectral radius of H,,-1 converges to that of H0,,-1 as goes to zero. Hence we can exploit the structure of H,,-1 as 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,,). Thirdly, If ==0, we consider the generalized operator Hp,m=ap(am+am)ap; (p,m=1,2,...) of -iH0,0, acting on Bargmann space. For this operator, we find some conditions on the parameters p and m for that Hp,m to be completely indeterminate. It follows from these conditions that Hp,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 Hp,m and Hp,m+Hp,m are connected at the chaotic operators (where Hp,m is the adjoint of the Hp,m).