Andrei and Wiegmann derived the exact solution to the s-d problem independently. If the s-d interaction is localized, only the radial degree of freedom needs to be considered, and the problem may be reduced to the Schrödinger equation in one-dimensional real space. When the solution is assumed to be in accord with Bethe's ansatz, the problem can be treated in exactly the same manner as in the one-dimensional Hubbard model, and the exact solution method used therein can be adapted as it is to the s–d problem. In this way, each of the various physical quantities can be represented by a single function all the way from high to low temperatures. This is a function of T/TK, and its functional form is found to be consistent with the result of Wilson, insofar as they can be compared. This approach is mathematically powerful, and we may apply it to the case with a magnetic field, the case with S >1/2, and the Anderson model. However, the focus of our attention in this chapter will be to discuss the initial analysis of Andrei (Andrei, 1980; Andrei and Lowenstein, 1981; Andrei et al., 1983) and Wiegmann (Weigmann, 1981; Filyov et al., 1981).

A one-dimensional model

At first, let us consider the movement of conduction electrons on a onedimensional line. Even in the three-dimensional case, if the interaction is δ function-like, only s-wave scattering arises.