Spectral theory of Jacobi operators with increasing coefficients. The critical case

Spectral properties of Jacobi operators J are intimately related to an asymptotic behavior of the corresponding orthogonal polynomials Pₙ(z) as n→∞ . We study the case where the off-diagonal coefficients aₙ and, eventually, diagonal coefficients bₙ of J tend to infinity in such a way that the ratio γₙ 2⁻¹bₙ(aₙaₙ₋₁)-1/2 has a finite limit γ . In the case |γ | < 1 asymptotic formulas for Pₙ(z) generalize those for the Hermite polynomials and the corresponding Jacobi operators J have absolutely continuous spectra covering the whole real line. If |γ | > 1 , then spectra of the operators J are discrete. Our goal is to investigate the critical case | γ |=1 that occurs, for example, for the Laguerre polynomials. The formulas obtained depend crucially on the rate of growth of the coefficients aₙ (or bₙ ) and are qualitatively different in the cases where aₙ→ ∞ faster or slower than n . For the fast growth of aₙ , we also have to distinguish the cases |γₙ| → 1-0 and |γₙ| → 1+0 . Spectral properties of the corresponding Jacobi operators are quite different in all these cases. Our approach works for an arbitrary power growth of the Jacobi coefficients.