Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space

In this paper, we present a “non-linear” factorization of a family of non-normal operators arising from Gribov’s theory of the following form: Hλ′,μ,λ=λ′A*2A2+μA*A+iλA*(A+A*)A, where the quartic Pomeron coupling λ′, the Pomeron intercept μ and the triple Pomeron coupling λ are real parameters, and i2=−1. A* and A are, respectively, the usual creation and annihilation operators of the one-dimensional harmonic oscillator obeying the canonical commutation relation [A,A*]=I. In Bargmann representation, we have A⟷ddz and A*⟷z, z=x+iy. It follows that Hλ′,μ,λ can be written in the following form: Hλ′,μ,λ=p(z)d2dz2+q(z)ddz, where p(z)=λ′z2+iλz and q(z)=iλz2+μz. This operator is an operator of the Heun type where the Heun operator is defined by H=p(z)d2dz2+q(z)ddz+v(z), where p(z) is a cubic complex polynomial, q(z) and v(z) are polynomials of degree at most 2 and 1, respectively, which are given. For z=−iy, Hλ′,μ,λ takes the following form: Hλ′,μ,λ=−a(y)d2dy2+b(y)ddz, with a(y)=y(λ−λ′y) and b(y)=y(λy+μ). We introduce the change of variable y=λ2λ′(1−cos(θ)), θ∈[0,π] to obtain the main result of transforming Hλ′,μ,λ into a product of two first-order operators: H˜λ′,μ,λ=λ′(ddθ+α(θ))(−ddθ+α(θ)), with α(θ) being explicitly determined.