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