On an application of generalized Jentzsch theorem to Gribov operator in Bargmann space

{\it In Bargmann representation, the reggeon’s field theory{\color{blue} [5]} is caracterized by the non symmetrical Gribov operator $\displaystyle{H_{\lambda’,\mu,\lambda} = \lambda’ A^{^{2}}A^{2} + \mu A^{}A + i\lambda A^{}(A + A^{})A}$where$A^{}$and$A$arethecreationandannihilationoperators;$[A, A^{}] = I $\mathrm{.}$(\lambda’,\mu, \lambda) \in $\mathbb{R}$^{3}$ are respectively the four coupling, the intercept and the triple coupling of Pomeron and $i^{2} = -1$. For ${\lambda }^{\mathrm{\prime }}>0,\mu >0$, let $\sigma ({\lambda }^{\mathrm{\prime }},\mu )\mathrm{\ne }0$ be the smallest eigenvalue of $H_{{\lambda }^{\mathrm{\prime }},\mu ,\lambda }$, we show in this paper that $\sigma ({\lambda }^{\mathrm{\prime }},\mu )$ is positive, increasing and analytic function on the whole real line with respect to $\mu$ and that the spectral radius of $H_{{\lambda }^{\mathrm{\prime }},\mu ,\lambda }^{-1}$ converges to that of $H_{0,\mu ,\lambda }^{-1}$ as ${\lambda }^{\mathrm{\prime }}$ goes to zero.\ The above results can be derived from the method used in ({\color{blue} [2]} Commun. Math. Phys. 93, (1984), p:123-139) by Ando-Zerner to study the smallest eigenvalue $\sigma (0,\mu )$ of $H_{0,\mu ,\lambda }$, however as $H_{{\lambda }^{\mathrm{\prime }},\mu ,\lambda }$ is regular perturbation of $H_{0,\mu ,\lambda }$ then its study is much more easily. We can exploit the structure of $H_{{\lambda }^{\mathrm{\prime }},\mu ,\lambda }^{-1}$ to deduce the results of Ando-Zerner established on the function $\sigma (0,\mu )$ as ${\lambda }^{\mathrm{\prime }}$ goes to zero.\}