Existence of Bound States in Continuous 0<D<\infty Dimensions

TL;DR

This paper investigates the existence of bound states in quantum systems across continuous dimensions from 0 to infinity, revealing that bound states always exist for dimensions up to 2 and require finite well sizes beyond that.

Contribution

It provides a comprehensive analysis of bound states in continuous D-dimensional quantum wells, including the conditions for their existence and the behavior as D varies.

Findings

Bound states always exist for 0<D≤2.

For D>2, finite well size is necessary for bound states.

Zero-energy bound states can be achieved with angular momentum tail.

Abstract

In modern fundamental theories there is consideration of higher dimensions, often in the context of what can be written as a Schr\"odinger equation. Thus, the energetics of bound states in different dimensions is of interest. By considering the quantum square well in continuous $D$ dimensions, it is shown that there is always a bound state for $0<D \le 2$. This binding is complete for D \to 0 and exponentially small for D \to 2_-. For D>2, a finite-sized well is always needed for there to be a bound state. This size grows like D^2 as D gets large. By adding the proper angular momentum tail a volcano, zero-energy, bound state can be obtained.