Density of the spectrum of Jacobi matrices with power asymptotics

We consider Jacobi matrices J whose parameters have the power asymptotics rho n = n beta 1 (x(0) + x(1/)n + O(n(-1-epsilon))) and q(n) = n(beta)2 (y(0) + y(1)/n + O(n(-1-epsilon))) for the off-diagonal and diagonal, respectively. We show that for beta(1) > beta(2), or beta(1) = beta(2) and 2x(0) > vertical bar y(0)vertical bar, 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 ->infinity) vertical bar q(n)|/rho(n) = 2) and determine the convergence exponent in almost all cases.