Dispersive Decay Estimates for periodic Jacobi operators on the half-line

TL;DR

This paper derives decay estimates over time for solutions of periodic Jacobi operators on the half-line, revealing specific decay rates depending on spectral properties and nondegeneracy conditions.

Contribution

It provides new dispersive decay estimates for periodic Jacobi operators, including explicit decay rates under various spectral conditions and nondegeneracy assumptions.

Findings

Proves $t^{-1/2}$ decay in weighted $ell^_{-1}$ norm.

Establishes $t^{-1/3}$ decay under a nondegeneracy condition.

Shows $t^{-1/(q+1)}$ decay for even period $q$ with $q$ disjoint spectral bands.

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.