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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 − 1 b n (a n a 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. [ABSTRACT FROM AUTHOR]