Attractive Interaction and Bound States

Abstract

Struggling to find a way to theoretically explain the phenomenon of superconductivity, in 1956 Cooper eventually reached a simplified version of the problem of two particles on the Fermi surface under a mutual attraction. Cooper’s problem, which represented a breakthrough in constructing a microscopic theory of superconductivity, is essentially identical to a one-particle problem with an attractive potential in two dimensions. In this chapter, we consider attractive potentials to clarify under what conditions a bound state is formed. First, we consider one-particle problems with an attractive potential in two and three dimensions to show that an infinitesimal attraction suffices in two dimensions to form a bound state whereas a finite threshold is requisite in three dimensions. Next, we shall see that this qualitative difference between two and three dimensions is caused by whether the one-particle density of states is finite at zero energy. Finally, the presence of the Fermi surface in Cooper’s problem will be shown to make the density of states at the excitation threshold finite even in three dimensions, resulting in the formation of a bound state from only an infinitesimal attraction.