Functional-Integral Approaches to the Anderson Model: A Comparison of Results in Limiting Cases

Functional-integral approaches to the Anderson model are checked and compared in simple limiting cases. We examine eight approximation schemes and no single scheme is found to perform correctly in all of the ten limiting cases. The two most sophisticated schemes we have investigated are those of Keiter and Hamann, respectively. The former suffers from deficiencies inherent in the 1-field functional-integral formalism (as described by Keiter), the latter from difficulties related to the extremal approximation to the $y$ field.