Pointwise convergence of solutions to the nonelliptic Schrodinger equation
Keith RogersAna VargasLuis Vega
35Q5542B25Schr\"odinger equationpointwise convergence
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.
Indiana University Mathematics Journal
2006
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10.1512/iumj.2006.55.2827
10.1512/iumj.2006.55.2827
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Indiana Univ. Math. J. 55 (2006) 1893 - 1906
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