Bound States of the One-Dimensional Maxwell--Schr\"odinger Equations

TL;DR

This paper proves the existence of ground states in the one-dimensional Maxwell--Schrödinger equations, introduces a new Banach space for the neutral atom case, and characterizes symmetry for point charges.

Contribution

It establishes ground state existence in 1D Maxwell--Schrödinger equations, introduces a novel Banach space for analysis, and describes symmetry properties of solutions.

Findings

Ground states exist for specified charge configurations.

A new quartic Banach space is introduced for the neutral atom case.

Ground states with point charges are symmetric and decreasing.

Abstract

We prove the existence of a ground state of the Maxwell--Schr\"odinger equations in one spatial dimension, describing a specified amount of free charge under the influence of a fixed charge. For one case (equal free and fixed charge, i.e., a neutral atom), we introduce a new type of quartic Banach space, in which the Hamiltonian is naturally coercive. We also show that for a point charge, for any ratio of charge such that the ground state exists, it is symmetric and decreasing.