Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $ \mathbb{R}\times\mathbb{T}^2 $

Abstract

<abstract><p>In the first part of the paper we continue the study of solutions to Schrödinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrödinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.</p></abstract>

On the uniqueness of variable coefficient Schrödinger equations

“…Note finally that the power R 2 (instead of R 3 of the general case) in the lower bound for δ(R) is allowed by the choice β = C 0 R 2 . See [17]. Since e ψ(y 1 ) ≤ e Notations.…”

On some variable coefficient Schrödinger operators on R × Rn and R × T 2

“…We now give the general statement of the Strichartz estimates proved in [11] on R × T d , with d ≥ 1, but later on we shall restrict ourselves to the case d = 2.…”

“…For our purposes, that is to solve SLIVPs, the following multilinear estimates proved in [11] are crucial. g (R) as in Definition 3.7.…”

“…Then, for every u 0 ∈ H s (T 2 ), there exists a unique solution of the IVP The proof of the previous theorems is based on the standard contraction argument in the Bourgain spaces Xs,b g (R×T 2 ) and on the use of the results in Proposition 3.9. For the proofs see [11]. Remark 3.13.…”

“…It is important to mention that in the Euclidean setting operators like L b are studied in the context of Bose-Einstein condensations and nonlinear optics, and that soliton solutions of different kind have been obtained depending on the time-dependent coefficients appearing in the equation (see [6,18,28,29]). The classes L g and L a 1 ,a 2 , instead, have been considered by the author and G. Staffilani in [11]. From the analysis of these classes it came to light that for some variable coefficient Shrödinger operators on R × T 2 sharp local-well-posedness results are still valid.…”