Existence of bound states in continuous 0< D

Abstract

In modern fundamental theories there is consideration of higher dimensions, often in the context of what can be written as a Schrödinger 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⩽2. This binding is complete for D→0 and exponentially small for D→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.