Title: Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients,
Abstract: We consider Jacobi matrices with eventually increasing sequences of diagonal
and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the
subordinate solution at the top of the essential spectrum, and the asymptotic
behavior of the spectral density at the top of the essential spectrum.
In particular, allowing on both diagonal and off-diagonal Jacobi parameters
perturbations of the free case of the form $- \sum_{j=1}^J c_j n^{-\tau_j} +
o(n^{-\tau_1-1})$ with $0 < \tau_1 < \tau_2 < \dots < \tau_J$ and $c_1>0$, we
find the asymptotic behavior of the $\log$ of spectral density to order
$O(\log(2-x))$ as $x$ approaches $2$.
Apart from its intrinsic interest, the above results also allow us to
describe the asymptotics of the spectral density for orthogonal polynomials on
the unit circle with real-valued Verblunsky coefficients of the same form.