Analytical Diagonalization of Fermi Gas-like Hamiltonians using the Sommerfeld-Watson Transformation

Abstract

The Sommerfeld-Watson transformation is a powerful mathematical technique widely used in physics to simplify summations over discrete quantum numbers by converting them into contour integrals in the complex plane. This method has applications in scattering theory, high-energy physics, quantum field theory, and electrostatics. A lesser-known but significant use is in the analytical diagonalization of specific Hamiltonians in condensed matter physics, such as the Fermi gas Hamiltonian and the single-impurity Anderson model with vanishing Coulomb repulsion. These models are used to describe important phenomena like conductance in metals, x-ray photoemission, and aspects of the Kondo problem. In this work, we provide a comprehensive explanation of the Sommerfeld-Watson transformation and its application in diagonalization procedures for these models, using modern notation to enhance clarity for new students. The analytical results were validated against the numerical diagonalization, showing excellent agreement. Furthermore, we extend the presented method to a more generalized non-interacting single-impurity Anderson model with variable couplings and arbitrary band dispersion. The procedure presented here successfully achieved the analytical diagonalization of this more complex model, providing a unified solution that encompasses simpler cases. To our knowledge, this general solution has not been previously reported.