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: H λ ′ , μ , λ = λ ′ A * 2 A 2 + μ A * A + i λ A * ( A + A * ) A , where the quartic Pomeron coupling λ ′ , the Pomeron intercept μ and the triple Pomeron coupling λ are real parameters, and i 2 = − 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 ⟷ d d z and A * ⟷ z , z = x + i y . It follows that H λ ′ , μ , λ can be written in the following form: H λ ′ , μ , λ = p ( z ) d 2 d z 2 + q ( z ) d d z , where p ( z ) = λ ′ z 2 + i λ z and q ( z ) = i λ z 2 + μ z . This operator is an operator of the Heun type where the Heun operator is defined by H = p ( z ) d 2 d z 2 + q ( z ) d d z + 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 = − i y , H λ ′ , μ , λ takes the following form: H λ ′ , μ , λ = − a ( y ) d 2 d y 2 + b ( y ) d d z , with a ( y ) = y ( λ − λ ′ y ) and b ( y ) = y ( λ y + μ ) . We introduce the change of variable y = λ 2 λ ′ ( 1 − c o s ( θ ) ) , θ ∈ [ 0 , π ] to obtain the main result of transforming H λ ′ , μ , λ into a product of two first-order operators: H ˜ λ ′ , μ , λ = λ ′ ( d d θ + α ( θ ) ) ( − d d θ + α ( θ ) ) , with α ( θ ) being explicitly determined.