The ℓp-boundedness of wave operators for the fourth-order Schrödinger operators on the lattice ℤ

Source abstract

This paper investigates the [Formula: see text]-boundedness of wave operators [Formula: see text] associated with discrete fourth-order Schrödinger operators [Formula: see text] on the lattice [Formula: see text], where [Formula: see text] and [Formula: see text] is a real-valued potential on [Formula: see text]. Under suitable decay assumptions on [Formula: see text] (depending on the types of zero resonance of [Formula: see text]), we show that the wave operators [Formula: see text] are bounded on [Formula: see text] for all [Formula: see text]: [Formula: see text] In particular, if both thresholds [Formula: see text] and [Formula: see text] are regular points of [Formula: see text], we prove that [Formula: see text] are neither bounded on the endpoint spaces [Formula: see text] nor on [Formula: see text]. We remark that the proof of these bounds relies fundamentally on the asymptotic expansions of the resolvent of [Formula: see text] near the two spectral thresholds, and on the theory of discrete singular integrals on the lattice. As an application, we derive the following [Formula: see text] decay estimates for solutions to the discrete beam equation with a parameter [Formula: see text] on the lattice [Formula: see text]: [Formula: see text] where [Formula: see text], [Formula: see text] is the conjugate exponent of [Formula: see text] and [Formula: see text] denotes the spectral projection onto the absolutely continuous spectral subspace of [Formula: see text].