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Schrieffer-Wolger Theory

A BCS theory variant that simplifies the treatment of superconductivity by neglecting the electron-electron interaction term.
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The statement of the theorem

Let the full Hamiltonian be H=H0+VH = H_0 + V, where H0H_0 is the unperturbed Hamiltonian and VV is the interaction term. The effective Hamiltonian HeffH_{eff} is obtained via second-order perturbation theory:\nHeff=H0+Veff=H0+V12(E1−E2)E1−E2V21+…\mathcal{H}_{eff} = H_0 + V_{eff} = H_0 + \frac{V_{12} (E_1 - E_2)}{E_1 - E_2} V_{21} + \dots \nFor the superconducting context, VeffV_{eff} describes the renormalized electron-electron interaction, effectively simplifying the treatment by projecting out high-energy degrees of freedom.