On Regularized Trace Formula of Gribov Semigroup Generated by the Hamiltonian of Reggeon Field Theory in Bargmann Representation

Abstract

In Intissar (J Math Anal Appl 305(2):669–689, 2005), the second author have considered the Gribov operator $${\mathbb {H}}_{\lambda '} = \lambda ' \mathbb {S} + {\mathbb {H}}_{\mu ,\lambda }$$ acting on Bargmann space where $$ \mathbb {S} = a^{*2} a^{2}$$ and $$ {\mathbb {H}}_{\mu ,\lambda } =\mu a^*a + i \lambda a^*(a+a^*)a$$ with $$i^{2} = -1$$ and $$(\lambda ^{'}, \mu , \lambda ) \in \mathbb {R}^{3}$$ . Here a and $$a^{*}$$ are the standard Bose annihilation and creation operators satisfying the commutation relation $$[a, a^{*}] = \mathbb {I}$$ . In Reggeon field theory, the real parameters $$\lambda {'}$$ is the four coupling of Pomeron, $$\mu $$ is Pomeron intercept, $$\lambda $$ is the triple coupling of Pomeron and $$i^{2} = -1$$ . He had given an approximation of the semigroup $$e^{-t{\mathbb {H}}_{\lambda '}}$$ generated by the operator $${\mathbb {H}}_{\lambda '}$$ . In particular, he had obtained an estimate approximation in trace norm of this semigroup by the unperturbed semigroup $$e^{-t\lambda '\mathbb {S}}$$ . In Intissar (J Math Anal Appl 437:59–70, 2016), he had regularized the operator $${\mathbb {H}}_{\mu ,\lambda }$$ by $$\lambda ''{\mathbb {G}}$$ where $${\mathbb {G}} = a^{*3} a^{3}$$ , i.e he had considered $${\mathbb {H}}_{\lambda ''} = \lambda '' {\mathbb {G}} + {\mathbb {H}}_{\mu ,\lambda }$$ where $$\lambda ''$$ is the magic coupling of Pomeron. In this case, he had established an exact relation between the degree of subordination of the non-self-adjoint perturbation operator $${\mathbb {H}}_{\mu ,\lambda }$$ to the unperturbed operator $${\mathbb {G}}$$ and the number of corrections necessary for the existence of finite formula of the regularized trace. The goal of the work of the authors in this article consists to study the trace of the semigroup $$e^{-t\mathbb {}H_{\lambda ''}}$$ , in particular to give an asymptotic expansion of this trace as $$t \rightarrow 0^{+}$$ .