Density of the spectrum of Jacobi matrices with power asymptotics

Abstract

We consider Jacobi matrices $J$ whose parameters have the power asymptotics ${\rho }_n=n^{{\beta }_1}\left(x_0+\frac{x_1}{n}+\mathrm{O}(n^{-1-\epsilon })\right)$ and $q_n=n^{{\beta }_2}\left(y_0+\frac{y_1}{n}+\mathrm{O}(n^{-1-\epsilon })\right)$ for the off-diagonal and diagonal, respectively. We show that for ${\beta }_1>{\beta }_2$, or ${\beta }_1={\beta }_2$ and $2x_0>|y_0|$, the matrix $J$ is in the limit circle case and the convergence exponent of its spectrum is $1/{\beta }_1$. Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix $J$ have a power asymptotic with one more term, we characterise the occurrence of the limit circle case completely (including the exceptional case ${\lim }_{n\to \infty }|q_n|/{\rho }_n=2$) and determine the convergence exponent in almost all cases.