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

Abstract

{\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 $A$ are the creation and annihilation operators; $[A, A^{*}] = I $.\\ $(\lambda',\mu, \lambda) \in \mathbb{R}^{3}$ are respectively the four coupling, the intercept and the triple coupling of Pomeron and $i^{2} = -1$. For $\lambda' > 0 ,\mu > 0$, let $\sigma (\lambda',\mu) \neq 0$ be the smallest eigenvalue of $H_{\lambda',\mu,\lambda}$, we show in this paper that $\sigma (\lambda',\mu)$ is positive, increasing and analytic function on the whole real line with respect to $\mu$ and that the spectral radius of $H_{\lambda',\mu,\lambda}^{-1}$ converges to that of $H_{0,\mu,\lambda}^{-1}$ as $\lambda'$ 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 $H_{\lambda',\mu,\lambda}$ is regular perturbation of $H_{0,\mu,\lambda}$ then its study is much more easily. We can exploit the structure of $H_{\lambda',\mu,\lambda}^{-1}$ to deduce the results of Ando-Zerner established on the function $\sigma (0,\mu)$ as $\lambda'$ goes to zero.\\}