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Spectral properties of Jacobi operators $J$ are intimately related to anasymptotic behavior of the corresponding orthogonal polynomials $P_{n}(z)$ as$n\to\infty$. We study the case where the off-diagonal coefficients $a_{n}$and, eventually, diagonal coefficients $ b_{n}$ of $J$ tend to infinity in sucha way that the ratio $\gamma_{n}:=2^{-1}b_{n} (a_{n}a_{n-1})^{-1/2} $ has afinite limit $ \gamma $. %We study an asymptotic behavior as $n\to\infty$ ofthe orthogonal polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients$a_{n}$ (off-diagonal terms) and $ b_{n}$ (diagonal terms). %We consider thecase $a_{n}\to\infty$ and suppose that the sequence $\gamma_{n}:=2^{-1}b_{n}(a_{n}a_{n-1})^{-1/2} $ has a limit $ \gamma $ as $n\to\infty$. In the case$|\gamma | < 1$ asymptotic formulas for $P_{n}(z)$ generalize those for theHermite polynomials and the corresponding Jacobi operators $J$ have absolutelycontinuous spectra covering the whole real line. If $|\gamma | > 1$, thenspectra of the operators $J$ are discrete. Our goal is to investigate thecritical case $| \gamma |=1$ that occurs, for example, for the Laguerrepolynomials. The formulas obtained depend crucially on the rate of growth ofthe coefficients $a_{n}$ (or $b_{n}$) and are qualitatively different in thecases where $a_{n}\to \infty$ faster or slower then $n$. For the fast growth of$a_{n}$, we also have to distinguish the cases  $|\gamma_{n}|  \to 1-0$ and $|\gamma_{n}|  \to 1+0$. Spectral properties of the corresponding Jacobi operators are quitedifferent in all these cases. Our approach works for an arbitrary power growthof the Jacobi coefficients.

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