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Note that this is a two-site problem, so the many-body Hilbert space is finite (and actually just four). Therefore one can simply express all the operators in terms of $4\times 4$ matrices. Let us order the many-body basis states as $|s_1\rangle=a_{1\uparrow}^\dagger a_{2\downarrow}^\dagger|\Omega\rangle$, \$|s_2\rangle=a_{2\uparrow}^\dagger ...

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I thinking a lot of time, then I came in this assertion: The kinetic part of the Anderson's hamiltonian have a continuous spectrum. We want to work numerically with this part, so we need to find a some basis that diagonalizes approximately (the terms off-diagonal be small by controlling some discretization parameter) in such a way that now the "spectrum" is ...

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Your question is deep and to the best of my knowledge only partially resolved, see Sec. II of 2007 article in Review of Modern Phsyics. The trick is called "the Wilson chain" and is essential for NRG to work on strongly correlated models with Kondo-like behavior. The Wilson chain construction makes NRG a computationally efficient procedure. My ...

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