Bound States in n Dimensions (Especially n = 1 and n = 2)

TL;DR

This paper investigates the existence and bounds of bound states in one and two-dimensional quantum systems, contrasting with higher dimensions, and provides examples and bounds that match expected behavior for large coupling constants.

Contribution

It demonstrates the absence of strict bounds on the number of bound states in low dimensions and derives bounds with correct coupling behavior for large coupling.

Findings

No strict bound on the number of bound states in 1D and 2D.

Examples of potentials with one or infinitely many bound states in 2D.

Derived bounds for 1D and 2D that match semi-classical behavior for large coupling.

Abstract

We stress that in contradiction with what happens in space dimensions $n \geq 3$, there is no strict bound on the number of bound states with the same structure as the semi-classical estimate for large coupling constant and give, in two dimensions, examples of weak potentials with one or infinitely many bound states. We derive bounds for one and two dimensions which have the "right" coupling constant behaviour for large coupling.