Scattering and Blow-up for threshold even solutions to the nonlinear Schr\"{o}dinger equation with repulsive delta potential at low frequencies

TL;DR

This paper investigates the long-term behavior of even solutions to a nonlinear Schrödinger equation with a delta potential, establishing a dichotomy between scattering and blow-up at low frequencies, including the critical frequency.

Contribution

It extends previous work by demonstrating a scattering and blow-up dichotomy for threshold solutions at low and critical frequencies in the presence of a delta potential.

Findings

Establishes scattering and blow-up behavior at low frequencies.

Confirms the dichotomy also holds at the critical frequency.

Provides a detailed analysis of threshold solutions with a delta potential.

Abstract

We consider the $L^2$-supercritical nonlinear Schr\"{o}dinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. In that case, there are obvious non-scattering global solutions, i.e., the standing waves. In this paper, we show a scattering and blow-up dichotomy for threshold even solutions in the low-frequency case. We emphasize that this dichotomy still holds at the critical frequency between high and low.