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

2009/06/19 by Peter Sjögren, Sjögren, Peter, J. L. Torrea +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0906.3642

openalex publication_date 2009/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

In the half-space ℝ^d × ℝ₊, we consider the Hermite-Schrödinger equation i∂ u/∂ t = - Δu + |x|² u, with given boundary values on 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.

Cited by

• Dispersive equations with invariant measures

Related

• Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces
• The nonlinear Schrödinger equation on the half-space
• Restriction theorem for the Fourier-Hermite transform and solution of the Hermite-Schrödinger equation
• The nonlinear Schrödinger equation in the half-space
• Pointwise convergence of the fractional Schrödinger equation in \mathbb R²