On regularized trace formula of Gribov semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann representation (Q1743872)

Suppose \({\mathbb E}\) is the Bargmann space of all entire functions \(f:{\mathbb C}\to {\mathbb C}\) such that \(\int_{\mathbb C}|f(z)|^2e^{-|z|^2}\,dx\,dy<\infty\). Let \({\mathbb H}_{\lambda''}=\lambda''{\mathbb G}+{\mathbb H}_{\mu,\lambda}\) be the Gribov operator acting on \({\mathbb E}\), where \({\mathbb G}=a^{*3}a^3\), \(a\) and its adjoint \(a^*\) are annihilation and creation operators, respectively, satisfying \([a,a^*]={\mathbb I}\), and \({\mathbb H}_{\mu,\lambda}=\mu a^*a+i\lambda a^*(a+a^*)a\). Here \(\lambda'',\mu,\lambda\) are real parameters and \(i^2=-1\). Let \(\|\cdot\|_1\) denote the trace-class norm. The main result of the paper is the following asymptotic formula: \(\big\|e^{-t{\mathbb H}_{\lambda''}}-e^{-t\lambda''{\mathbb G}}\big\|_1= t\big\|e^{-t\lambda''{\mathbb G}}{\mathbb H}_{\mu,\lambda}\big\|_1+ \big\| (\lambda''{\mathbb G})^{\delta} e^{-\frac{t}{3}\lambda''{\mathbb G}} \big\|_1 O(t^2)\) as \(t\to 0^+\) for \(\delta\geq 1/2\).