Density of the spectrum of Jacobi matrices with power asymptotics

Abstract

We consider Jacobi matrices J whose parameters have the power asymptotics ${\mathit{\rho }}_{\mathit{n}}={\mathit{n}}^{{\mathit{\beta }}_1}\mathrm{(}{\mathit{x}}_0+\frac{{\mathit{x}}_1}{\mathit{n}}+\mathrm{O}\mathrm{(}{\mathit{n}}^{-1-\mathit{ϵ}}\mathrm{)}\mathrm{)}$ and ${\mathit{q}}_{\mathit{n}}={\mathit{n}}^{{\mathit{\beta }}_2}\mathrm{(}{\mathit{y}}_0+\frac{{\mathit{y}}_1}{\mathit{n}}+\mathrm{O}\mathrm{(}{\mathit{n}}^{-1-\mathit{ϵ}}\mathrm{)}\mathrm{)}$ for the off-diagonal and diagonal, respectively.

We show that for ${\mathit{\beta }}_1\mathrm{>}{\mathit{\beta }}_2$ , or ${\mathit{\beta }}_1={\mathit{\beta }}_2$ and $2{\mathit{x}}_0\mathrm{>}|{\mathit{y}}_0|$ , the matrix J is in the limit circle case and the convergence exponent of its spectrum is $1\mathrm{/}{\mathit{\beta }}_1$ . 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 ${lim}_{\mathit{n}\to \infty }|{\mathit{q}}_{\mathit{n}}|\mathrm{/}{\mathit{\rho }}_{\mathit{n}}=2$ ) and determine the convergence exponent in almost all cases.