Dispersive estimates for solutions to the perturbed one-dimensional Klein-Gordon equation with and without a one-gap periodic potential (Q2929414)

The potential \(V(x)\) in the considered one-dimensional linear Klein-Gordon equation is the sum of a special periodic potential, a Lamé potential, and a spatially localized perturbation. The aim of the paper is to prove some decay estimates for the solution of the attached Cauchy problem. Under some special assumptions on initial data and potential, three time-decay rates for the solutions are proved. In case the spatially localized potential is ``generic'', the time-decay rate involves the weighted \( L^2\) norm of solution. In case one excludes a finite number of masses \(\mu\) , the \( L^{\infty}\) norm of solution is estimated. In case spatially periodic potential is zero and the localized potential is ``generic'', still \( L^{\infty}\) norm of solution is estimated. In all cases , the estimates involve weighted Sobolev norms of the initial conditions. A collection of known results that are useful for understanding the proofs is presented in the sections ``Spectral properties and integral transforms'' and Band functions and distorted Bloch waves. Sections 4--6 contain the proof of all three estimations. Successively the discussion covers the weighted decay rate of \( t^{-3/2}\) , uniform decay rate of \( t^{-1/3}\) and improved uniform decay rate of \( t^{-1/2}\).