The Cauchy problem for 1D nonlinear Schrödinger equations with repulsive delta potential for data in $L^p$-based spaces

Qingquan Deng, Ping Li and Xiuhong Long

Vol. 342 (2026), No. 1, 115–161

DOI: 10.2140/pjm.2026.342.115

Abstract

Let $\lambda \in ℝ$ and let $H=-\frac{1}{2}{\partial }_x^2+q{\delta }_0$ be the one-dimensional Schrödinger equation with a repulsive delta potential. We study the Cauchy problem for the nonlinear equation

$$\{\begin{array}{c}i{\partial }_{t}u(t,x)=Hu(t,x)+\lambda |u(t,x){|}^{2}u(t,x),(t,x)\in ℝ\times ℝ,\hfill \\ u(0,x)=u_0(x),\hfill \end{array}$$

in $L^p$-based spaces. Using the boundedness of wave operators, a characterization of Besov space adapted to $H$, and the cancellation property of the trilinear form $\mathcal{𝒯}(v_1(\tau ),v_2(\tau ),v_3(\tau ))=\mathcal{𝒰}(-\tau )\left(\mathcal{𝒰}(-\tau )v_1(\tau )\mathcal{𝒰}(\tau )v_2(\tau )\mathcal{𝒰}(\tau )v_3(\tau )\right)$ with $\mathcal{𝒰}(\tau )=e^{-itH}$, we demonstrate that under the linear transformation $v(t)=\mathcal{𝒰}(-t)u(t)$, the problem is locally well-posed in $L^p(ℝ)$ for $1<p<2$ and in the homogeneous Besov space ${Ḃ}_{p,1}^s(ℝ)$ with $1<p<2$ and $s=1-\frac{1}{p}$.

Keywords

Cauchy problem, nonlinear Schrödinger equation, wave operator, locally well-posedness, delta potential

Mathematical Subject Classification

Primary: 35Q55, 35A01, 42A38, 42B35

Milestones

Received: 2 November 2024

Revised: 24 June 2025

Accepted: 4 January 2026

Published: 23 March 2026