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

Tom 118 / 2010

Peter Sjögren, J. L. Torrea 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
    Facultad de Ciencias
    Universidad Autónoma de Madrid
    28049 Madrid, Spain
    e-mail

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