# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## On the boundary convergence of solutions to the Hermite–Schrödinger equation

### Tom 118 / 2010

Colloquium Mathematicum 118 (2010), 161-174 MSC: Primary 42B35, 42B37; Secondary 42C10, 35K15. DOI: 10.4064/cm118-1-8

#### Streszczenie

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.

#### Autorzy

• Peter SjögrenMathematical Sciences
University of Gothenburg
SE-412 96 Göteborg, Sweden
and
Mathematical Sciences
Chalmers
SE-412 96 Göteborg, Sweden
e-mail
• J. L. TorreaDepartamento de Matemáticas
and ICMAT CSIC-UAM-UCM-UC3M