Properties of groundstates of nonlinear Schr\"odinger equations under a weak constant magnetic field

TL;DR

This paper investigates the properties of groundstates for a nonlinear Schrödinger equation with a weak constant magnetic field, revealing uniqueness, symmetry, decay behavior, and energy convexity in this regime.

Contribution

It provides new insights into the qualitative behavior of groundstates under small magnetic fields, including uniqueness, symmetry, decay, and energy properties.

Findings

Groundstates are unique up to magnetic translations and rotations.

Groundstates exhibit rotational invariance matching the magnetic field.

Magnetic fields induce Gaussian decay in groundstates.

Abstract

We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schr\"odinger equation \[ - (\nabla + i A)^2 u + u = |u|^{p-2} u, \qquad \text{ in } \mathbb{R}^N, \] where the magnetic potential $A$ induces a constant magnetic field. When the latter magnetic field is small enough, we show that the groundstate solution is unique up to magnetic translations and rotations in the complex phase space, that groundstate solutions share the rotational invariance of the magnetic field and that the presence of a magnetic field induces a Gaussian decay. In this small magnetic field r\'egime, the corresponding ground-energy is a convex differentiable function of the magnetic field.