Factorization of the Non-Normal Hamiltonian of Reggeon Field Theory in Bargmann Space - Abdelkader Intissar (Mathematics)
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: <semantics><mstyle scriptlevel="0" displaystyle="true">Hλ′,μ,λ=λ′A*2A2+μA*A+iλA*</mstyle></semantics><semantics>(A+A*)A,</semantics> where the quartic Pomeron coupling <semantics>λ′</semantics>, the Pomeron intercept <semantics>μ</semantics> and the triple Pomeron coupling <semantics>λ</semantics> are real parameters, and <semantics>i2=−1</semantics>. <semantics>A*</semantics> and A are, respectively, the usual creation and annihilation operators of the one-dimensional harmonic oscillator obeying the canonical commutation relation <semantics>[A,A*]=I</semantics>. In Bargmann representation, we have <semantics><mstyle scriptlevel="0" displaystyle="true">A⟷ddz</mstyle></semantics> and <semantics><mstyle scriptlevel="0" displaystyle="true">A*⟷z</mstyle></semantics>, <semantics>z=x+iy</semantics>. It follows that <semantics><mstyle scriptlevel="0" displaystyle="true">Hλ′,μ,λ</mstyle></semantics> can be written in the following form: <semantics><mstyle scriptlevel="0" displaystyle="true">Hλ′,μ,λ=p(z)d2dz2+q(z)ddz,</mstyle></semantics> where <semantics>p(z)=λ′z2+iλz</semantics> and <semantics>q(z)=iλz2+μz.</semantics> This operator is an operator of the Heun type where the Heun operator is defined by <semantics>H=p(z)<mstyle scriptlevel="0" displaystyle="true">d2dz2</mstyle>+q(z)<mstyle scriptlevel="0" displaystyle="true">ddz</mstyle>+v(z),</semantics> where <semantics>p(z)</semantics> is a cubic complex polynomial, <semantics>q(z)</semantics> and <semantics>v(z)</semantics> are polynomials of degree at most 2 and 1, respectively, which are given. For <semantics>z=−iy</semantics>, <semantics>Hλ′,μ,λ</semantics> takes the following form: <semantics>Hλ′,μ,λ=−a(y)<mstyle scriptlevel="0" displaystyle="true">d2dy2</mstyle>+b(y)<mstyle scriptlevel="0" displaystyle="true">ddz</mstyle>,</semantics> with <semantics>a(y)=y(λ−λ′y)</semantics> and <semantics><mstyle scriptlevel="0" displaystyle="true">b(y)=y(λy+μ).</mstyle></semantics> We introduce the change of variable <semantics>y=<mstyle scriptlevel="0" displaystyle="true">λ2λ′</mstyle>(1−cos(θ))</semantics>, <semantics>θ∈[0,π]</semantics> to obtain the main result of transforming <semantics><mstyle scriptlevel="0" displaystyle="true">Hλ′,μ,λ</mstyle></semantics> into a product of two first-order operators: <semantics><mstyle scriptlevel="0" displaystyle="true"><mover accent="true">H˜</mover>λ′,μ,λ=λ′(ddθ+α(θ))(−ddθ+α(θ)),</mstyle></semantics> with <semantics>α(θ)</semantics> being explicitly determined.