IUMJ

Title: Pointwise convergence of solutions to the nonelliptic Schrodinger equation

Authors: Keith M. Rogers, Ana Vargas and Luis Vega

Issue: Volume 55 (2006), Issue 6, 1893-1906

Abstract:

It is conjectured that the solution to the Schr\"odinger equation in $\mathbb{R}^{n+1}$ converges almost everywhere to its initial datum $f$, for all $f \in H^{s}(\mathbb{R}^n)$, if and only if $s \ge \frac{1}{4}$. It is known that there is an $s < \frac{1}{2}$ for which the solution converges for all $f \in H^{s}(\mathbb{R}^{2})$. We show that the solution to the nonelliptic Schr\"odinger equation, $i\partial_{t}u + (\partial^{2}_{x} - \partial^{2}_{y})u = 0$, converges to its initial datum $f$, for all $f \in H^{s}(\mathbb{R}^{2})$, if and only if $s \ge \frac{1}{2}$. Thus the pointwise behaviour is worse than that of the standard Schr\"odinger equation. In higher dimensions, we have similar results with the loss of the endpoint.