Dispersive Decay Estimates for periodic Jacobi operators on the half-line
Amir Sagiv, Remy Kassem, Michael I Weinstein
TL;DR
The paper analyzes dispersive time-decay for the discrete Schrödinger equation on the half-line with a periodic Jacobi operator $J$, deriving sharp decay rates via spectral-band analysis and oscillatory-integral bounds. It constructs the propagator through an eigenfunction expansion tied to the band structure, and reduces the problem to oscillatory integrals with phase functions $k_j$ arising from the discriminant data. The main contributions are a local decay bound $\|e^{-itJ}P_c\|_{\ell^1_1\to\ell^\infty_{-1}}\lesssim t^{-1/2}$ for all periodic $J$, and global bounds $\|e^{-itJ}P_c\|_{\ell^1\to\ell^\infty}\lesssim t^{-1/3}$ under a nondegeneracy condition on the phase, or $\lesssim t^{-1/(q+1)}$ when the spectrum splits into exactly $q$ bands with even $q$. These results illuminate edge-transport and radiation-damping phenomena in one-dimensional Floquet-like systems and provide a framework for extensions to nonlinear or driven discrete media.
Abstract
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 $\ell^\infty_{-1}$ 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 $q\geq2$, if the continuous spectrum consists of exactly $q$ disjoint intervals (bands), we obtain a $t^{-1/(q+1)}$ decay rate without any further assumptions.