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 $P_n(z)$ as $n\to \mathrm{\infty }$. We study the case where the off-diagonal coefficients $a_n$ and, eventually, diagonal coefficients $b_n$ of $J$ tend to infinity in such a way that the ratio ${\gamma }_n:=2^{-1}{b}_n(a_n{a}_{n-1}{)}^{-1\mathrm{/}2}$ has a finite limit $\gamma$. %We study an asymptotic behavior as $n\to \mathrm{\infty }$ 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 $a_n\to \mathrm{\infty }$ and suppose that the sequence ${\gamma }_n:=2^{-1}{b}_n(a_n{a}_{n-1}{)}^{-1\mathrm{/}2}$ has a limit $\gamma$ as $n\to \mathrm{\infty }$. In the case $\mathrm{\mid }\gamma \mathrm{\mid }<1$ asymptotic formulas for $P_n(z)$ generalize those for the Hermite polynomials and the corresponding Jacobi operators $J$ have absolutely continuous spectra covering the whole real line. If $\mathrm{\mid }\gamma \mathrm{\mid }>1$, then spectra of the operators $J$ are discrete. Our goal is to investigate the critical case $\mathrm{\mid }\gamma \mathrm{\mid }=1$ that occurs, for example, for the Laguerre polynomials. The formulas obtained depend crucially on the rate of growth of the coefficients $a_n$ (or $b_n$) and are qualitatively different in the cases where $a_n\to \mathrm{\infty }$ faster or slower then $n$. For the fast growth of $a_n$, we also have to distinguish the cases $\mathrm{\mid }{\gamma }_n\mathrm{\mid }\to 1-0$ and $\mathrm{\mid }{\gamma }_n\mathrm{\mid }\to 1+0$. 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.