Sharp Strichartz estimates on quadratic irrational tori

Abstract

We prove sharp $L^{4}$ Strichartz estimates for two-dimensional periodic Schr\"{o}dinger flows with integral diagonal symbols $ak_{1}^2+bk_2^2$ where $a,b \in \mathbb Z \setminus \{0\}$. The estimates exhibit a sharp arithmetic dichotomy according to whether $-ab$ is a square: for frequencies bounded by $N$, the optimal loss is $(\log N)^{1/4}$ when $-ab$ is not a square and $N^{1/4}$ when $-ab$ is a square. As a consequence, we show that arbitrarily small perturbations of the rectangular aspect ratio can lead to a transition between global well-posedness and norm inflation for cubic hyperbolic NLS.