Bounds states of the Schr\"odinger-Newton model in low dimensions

TL;DR

This paper proves the existence and uniqueness of symmetric solutions with specific zeros for the Schrödinger-Newton model in low dimensions, using differential equation analysis.

Contribution

It establishes the existence and uniqueness of symmetric solutions with prescribed zeros for the Schrödinger-Newton model in one and two dimensions.

Findings

Existence of solutions with exactly n zeros for n≥0.

Uniqueness of the zero-zero solution in the one-dimensional case.

Analysis of differential equations underpinning the solutions.

Abstract

We prove the existence of quasi-stationary symmetric solutions with exactly n>=0 zeros and uniqueness for n=0 for the Schr\"odinger-Newton model in one dimension and in two dimensions along with an angular momentum m>=0. Our result is based on an analysis of the corresponding system of second-order differential equations.