Zoran Skoda Fano-Anderson model

This model was discovered simultaneously in atomic physics by Fano and in condensed matter physics by Anderson.

It involves a continuous spectrum of free particles with energies $\epsilon_k$ and number operator $n_k = a_k^\dagger a_k$ and a single extra particle in state $|v\rangle$ and energy $\epsilon_v$. The Hamiltonian in this model is of the form

$$H = \sum_k \epsilon_k n_k + \epsilon_v n_v + \sum_k (V_k a_k^\dagger a_v + \bar{V}_k a_v^\dagger a_k)$$

The complex coefficients $V_k$ and $\bar{V}_k$ are called the hopping matrix elements as they encode the hopping transitions between the single particle state and continuum.

• Ugo Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev. 124, 1866 (1961) doi
• P.W. Anderson, Phys. Rev.164, 41 (1961)
• G.D. Mahan, Many-particle physics (New York, Plenum Press, 1990) 272–285
• Michele Cini, Topics and methods in condensed matter theory, 2007

It is related also to the classical theory of

• J. von Neumann, E. Wigner, Phys. Z. 30, 465 (1929)