The Sommerfeld-Watson transformation in teaching condensed matter physics: analytical diagonalization of Fermi gas-like models

This work presents the application of the lesser-known yet significant Sommerfeld-Watson transformation (SWT) for the analytical diagonalization of specific Hamiltonians in condensed matter physics. This powerful mathematical technique is particularly useful for simplifying summations over discrete quantum numbers by converting them into contour integrals in the complex plane. We focus on the Fermi gas and the non-interacting single-impurity Anderson model, both fundamental for describing important phenomena such as electronic conductance in metals, x-ray photoemission, and aspects of the Kondo problem. A comprehensive explanation of the SWT is provided, and its role in the diagonalization of these models is illustrated using modern notation for enhanced clarity. The analytical results obtained were validated against direct numerical diagonalization, showing excellent agreement. Additionally, we extended the method to a generalized version of the non-interacting single-impurity Anderson model, incorporating variable couplings and an arbitrary band dispersion, and provided two non-trivial examples. The procedure successfully achieved the analytical diagonalization of this more complex model, offering a generalized solution not previously reported in the literature. The detailed presentation, modern notation, and numerical validation of the SWT applied to key models in condensed matter physics constitute a valuable pedagogical resource, enriching the teaching of theoretical condensed matter physics.