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In this chapter, the time-independent Green’s functions are defined, their main properties are presented, methods for their calculation are briefly discussed, and their use in problems of physical interest is summarized.

The Green’s functions corresponding to linear partial differential equations of first and second order in time are defined; their main properties and uses are presented.

The general theory developed in Chap. 1 can be applied directly to the time-independent one-particle Schrödinger equation by making the substitutions ()→ℌ(), λ → , where ℌ() is the Hamiltonian. The formalism presented in Chap. 2, Sects. 2.1,2.2 is applicable to the time-dependent one-particle Schrödinger equation.

The problem of finding the eigenvalues and eigenfunctions of a Hamiltonian ℌ = ℌ + ℌ can be solved in three steps: 1) Calculate the Green’s function () corresponding to ℌ. 2) Express () as a perturbation series in terms of () and ℌ, where () is the Green’s function associated with ℌ. 3) Extract from () information about the eigenvalues and eigenfunctions of ℌ.

We introduce the so-called tight-binding Hamiltonians (TBH), which have the form where each state ∣ι〉 is an atomiclike orbital centered at the site ι; the sites {ι} form a lattice. Such Hamiltonians are very important in solid-state physics. Here we calculate the Green’s functions associated with the TBH for various simple lattices. We also review briefly some applications in solid-state physics.

In this chapter we examine (using techniques developed in Chap. 4) a model tight-binding Hamiltonian describing the problem of a substitutional impurity in a perfect periodic lattice. We obtain explicit results for bound and scattering states. Certain important applications, such as gap levels in solids, Cooper pairs in superconductivity, resonance and bound states producing the Kondo effect, and impurity lattice vibrations, are presented.

In this chapter we examine first a system consisting of two “impurities” embedded in a periodic tight-binding model (TBM). This prepares the way for the approximate treatment of a disordered system containing a nonzero concentration of impurities. We examine in particular the average DOS, which depends on the average Green’s function, 〈〉. For the calculation of the latter, various approximation methods are employed; prominent among them is the so-called coherent potential approximation (CPA).

Disorder has a much more pronounced effect on transport properties than on the DOS. In fact, the DC metallic electrical conductivity is finite and not infinite (at = 0), because of the presence of disorder, no matter how weak. As the disorder increases further, it may produce a metal-insulator transition, i.e., it may prevent the propagation of the carriers altogether, making the conductivity equal to zero (at = 0 K). In the last 30 years there have been impressive developments in our understanding of these phenomena and in elucidating the role of elastic and inelastic scattering; Green’s functions have played a central role as a theoretical tool. In this chapter, we shall introduce several transport quantities, such as electrical conductivity, and present several schemes for their calculation.

The Green’s functions defined in Chap. 10 have analytical properties that are similar but not identical to the Green’s functions defined in Chap. 2 corresponding to second-order (in time) differential equations. They can all be expressed in terms of a generalized DOS and the Fermi or Bose thermal equilibrium distributions. From the Green’s functions (or the generalized DOS) one can easily obtain all thermodynamic quantities and linear response functions like the conductivity. The poles of an appropriate analytic continuation of in the complex -plane can be interpreted as the energy (the real part of the pole) and the inverse lifetime (the imaginary part of the pole) of quasiparticles. The latter are entities that allow us to map an interacting system to a noninteracting one.

The Green’s functions defined earlier are recast in a second quantized form. The resulting expressions can easily be generalized for the case where there are many particles. The time evolution of the operators involves now the interaction terms in the Hamiltonian. As a result, the generalized Green’s functions obey differential equations containing extra terms that depend on more complicated Green’s functions.