On the boundary convergence of solutions to the Hermite–Schrödinger equation
In the half-space $\mathbb R^d \times \mathbb R_+$, consider the Hermite–Schrödinger 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 Schrödinger equation. It shows that mixed norm estimates for the Hermite–Schrödinger 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.