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Approximation of the semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann space. (English) Zbl 1072.47038

Let \(A\) and \(A^*\) be the standard annihilation and creation operators in the Bargmann space. A non-self adjoint Gribov operator is defined by \[ H_{\lambda',\mu}= \lambda' A^{*2} A^2 + \mu A^* A + i \lambda A^*(A^*+A)A, \] where \(\mu\) is the Pomeron intercept, \(\lambda'\) is a four Pomeron coupling and \(i\lambda \) an imaginary Pomeron coupling. The operator \(H_{\lambda',\mu}\) governs the Reggeon field theory.
The paper under review studies the dynamic defined by \(H_{\lambda',\mu}\) through the Trotter product formula. To this end, \(H_{\lambda',\mu}\) is split into sums of either \(\lambda' A^{*2} A^2\) and \(\mu A^* A + i \lambda A^*(A^*+A)A\) or \(\lambda' A^{*2} A^2 + \mu A^* A\) and \( i \lambda A^*(A^*+A)A\). For the first case, a specific error estimation is proved. In the second case, an approximation of the dynamic semigroup is obtained. The standard Trotter formula does not work here since \(A^*(A^*+A)A\) is not the infinitesimal generator of a semigroup. The paper contains a new technique which can handle such an obstacle.

MSC:

47D06 One-parameter semigroups and linear evolution equations
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N50 Applications of operator theory in the physical sciences
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
81V05 Strong interaction, including quantum chromodynamics
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