Ill-posedness conjecture for the two-dimensional Schrödinger equation with an $L^1+L^\infty$ potential

Let $d=2$ and consider the nonlinear Schrödinger equation (eq:NLS) with potential $\eta \in L_x^1+L_x^{\infty}(\mathbb{R}^2)$. For $\gamma \in \mathbb{R}$, well-posedness in $H_x^\gamma(\mathbb{R}^2)$ means existence of a solution theory with the usual uniqueness and continuous-dependence properties, while ill-posedness means that this well-posedness fails.

Ill-posedness conjecture. For any $\gamma\in \mathbb{R}$, there exists $\eta \in L_x^1+L_x^{\infty}(\mathbb{R}^2)$ such that (eq:NLS) is ill-posed in $H_x^\gamma(\mathbb{R}^2)$.

The preceding results establish sharp well-posedness in all other parameter regimes considered in the paper, leaving only the case $d=2$ and $r=1$. The conjecture asserts that no Sobolev regularity yields well-posedness for at least one potential in this endpoint class.