On regularized trace formula of Gribov semigroup genrated by the Hamiltonian of reggeon field theory in Bargmann representation

Abstract

In J. Math. Anal. App. 305. (2005), we have considered the Gribov operator\\ $H_{λ'} = λ' S + H_{μ,λ}$ acting on Bargmann space where $ S = a^{*2} a^{2}$ and $ H_{μ,λ} =μa^* a + \ıλa^* (a+a^*)a$ with $i^{2} = -1$.\\ Here $a$ and $a^{*}$ are the standard Bose annihilation and creation operators satisfying the commutation relation $[a, a^{*}] = I$. In Reggeon field theory, the real parameters $λ{'}$ is the four coupling of Pomeron, $μ$ is Pomeron intercept, $λ$ is the triple coupling of Pomeron and $i^{2} = -1$.\\ We have given an approximation of the semigroup $e^{-tH_{λ'}}$ generated by the operator $H_{λ'}$. In particulary, we have obtained an estimate approximation in trace norm of this semigroup by the unperturbed semigroup $e^{-tλ'S}$. In {\bf[12]}, we have regularized the operator $H_{μ,λ}$ by $λ''G$ where $G = a^{*3} a^{3}$, i.e we have considered $H_{λ''} = λ'' G + H_{μ,λ}$ where $λ''$ is the {\it magic coupling} of Pomeron. In this case, we have established an exact relation between the degree of subordination of the non-self-adjoint perturbation operator $H_{μ,λ}$ to the unperturbed operator $G$ and the number of corrections necessary for the existence of finite formula of the regularized trace. The goal of this article consists to study the trace of the semigroup $e^{-tH_{λ''}}$, in particular to give an asymptotic expansion of this trace as $t \rightarrow 0^{+}$.