On the zeros of some families of polynomials satisfying a three-term recurrence associated to Gribov operator

Abstract

We consider families of tridiagonal- matrices with diagonal $\beta_{k} = \mu k$ and off-diagonal entries $\alpha_{k} = i\lambda k\sqrt{k+1}$; $1 \leq k \leq n$, $n \in \mathbb{N}$ and $i^{2} = -1$ where $\mu \in \mathbb{C}$ and $\lambda \in \mathbb{C}$.\\\quad In Gribov theory ([7], A reggeon diagram technique, Soviet Phys. JETP 26 (1968), no. 2, 414-423), the parmeters $\mu$ and $\lambda$ are reals and they are important in the reggeon field theory. In this theory $\mu$ is the intercept of Pomeron which describes the energy of dependence of total hadronic cross sections in the currently available range of energies and $\lambda$ is the triple coupling of Pomeron. The main motive of the paper is the localization of eigenvalues $z_{k,n}(\mu, \lambda)$ of the above matrices which are the zeros of the polynomials $P_{n+1}^{^{\mu,\lambda}}(z)$ satisfying a three-term recurrence : $\left\{\begin{array}[c]{l}P_{0}^{^{\mu,\lambda}}(z) = 0\\\quad\\ P_{1}^{^{\mu,\lambda}}(z) = 1\\\quad \\ \alpha_{n-1}P_{n-1}^{^{\mu,\lambda}}(z) + \beta_{n}P_{n}^{^{\mu,\lambda}}(z) + \alpha_{n}P_{n+1}^{^{\mu,\lambda}}(z) = zP_{n}^{^{\mu,\lambda}}(z);\quad n\geq 1\\ \end{array} \right. $ \quad \n If $\mu \in \mathbb{R}$ and $\lambda \in \mathbb{R}$ then the above matrices are complex symmetric, in this case we show existence of complex-valued function $\xi(z)$ of bounded variation on $\mathbb{R}$ such that the polynomials $P_{n}^{^{\mu,\lambda}}(z)$ are orthogonal with this weight $\xi(z)$.\\ }