Bound States in one and two Spatial Dimensions

TL;DR

This paper investigates the number of bound states in one and two-dimensional quantum potentials, providing bounds and examples of potentials with infinitely many bound states under certain conditions.

Contribution

It introduces explicit bounds on the number of bound states in 1D and 2D, and constructs examples with infinitely many bound states for weak potentials.

Findings

Weak attractive potentials can have infinitely many bound states.

Explicit bounds are derived for the number of bound states in 1D and 2D.

Conditions under which the number of bound states is finite or infinite are clarified.

Abstract

In this paper we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the well-known fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the three-dimensional zero angular momentum. A change of variables which allows us to go from the one-dimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the non-central case.