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arXiv · 2608.11834

Scattering Theory For 3D Cubic Damped Magnetic Schrödinger Equation

Luca Fanelli·Haruya Mizutani·Yilin Song·Ying Wang·Jiqiang Zheng
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Abstract

We consider the three-dimensional defocusing cubic nonlinear Schrödinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schrödinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.

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Luca Fanelli, Haruya Mizutani, Yilin Song, Ying Wang, Jiqiang Zheng. 2026-08-12. Scattering Theory For 3D Cubic Damped Magnetic Schrödinger Equation. https://arxiv.org/abs/2608.11834

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