Dispersive decay estimates for periodic Jacobi operators on the half-line

We establish dispersive time-decay estimates for periodic Jacobi operators on the discrete half-line, $N$. Specifically, we prove $t^{-1/2}$ decay in the weighted ${\ell }_{-1}^{\infty }$ norm for all such operators. For the global ${\ell }^1\to {\ell }^{\infty }$ decay estimate, we show that $t^{-1/3}$ decay holds under a nondegeneracy condition on the discriminant. Alternatively, for any even period $q\ge 2$, if the continuous spectrum consists of exactly q disjoint intervals (bands), we obtain a $t^{-1/(q+1)}$ decay rate without any further assumptions.