Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials P _n(z) defined by Jacobi recurrence coefficients a _n (off-diagonal terms) and b _n (diagonal terms). We consider the case a _n→∞, b _n→∞ in such a way that ∑a _n ⁻¹<∞ (that is, the Carleman condition is violated) and γ _n:=2 ⁻¹b _n(a _na _n−1) ^−1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for P _n(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γ _n|→1−0 and |γ _n|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P _n(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.