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

Abstract

Spectral properties of Jacobi operators $J$ are intimately related to an asymptotic 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 such a way that the ratio ${\gamma }_n:-2^{-1}{b}_n(a_n{a}_{n-1}{)}^{-1/2}$ has a finite limit $\gamma$. In the case $|\gamma |<1$ 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 $|\gamma |>1$, then spectra of the operators $J$ are discrete. Our goal is to investigate the critical case $|\gamma |=1$ 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 $a_n\to \infty$ faster or slower than $n$. 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 quite different in all these cases. Our approach works for an arbitrary power growth of the Jacobi coefficients.