On the boundary convergence of solutions to the Hermite-Schrödinger equation

Details On the boundary convergence of solutions to the Hermite-Schrödinger equation On the boundary convergence of solutions to the Hermite-Schrödinger equation

2009-06-19

arxiv.org/abs/0906.3642v1

Mathematics

Analysis of PDEs

12 pages

Scientific paper

In the half-space $\mathbb{R}^d \times \mathbb{R}_+$, we consider the Hermite-Schr\"odinger equation $i\partial u/\partial t = - \Delta u + |x|^2 u$, with given boundary values on $\mathbb{R}^d$. We prove a formula that links the solution of this problem to that of the classical Schr\"odinger equation. It shows that mixed norm estimates for the Hermite-Schr\"odinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link.

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