Greens Functions in Quantum Physics by Eleftherios N. Economou

By Eleftherios N. Economou

The most a part of this ebook is dedicated to the easiest type of Green's features, specifically the options of linear differential equations with a -function resource. it really is proven that those time-honored Green's services are a robust software for acquiring particularly easy and common options of simple difficulties comparable to scattering and boundlevel info. The bound-level remedy offers a transparent actual knowing of "difficult" questions equivalent to superconductivity, the Kondo impact, and, to a lesser measure, disorder-induced localization. The extra complicated topic of many-body Green's features is gifted within the final a part of the ebook.

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2d can be expressed in terms of the other three because gR + gA = g + g− , as can be seen from Fig. 2. 2 Second-Order Case ω-plane (a) g ③ ✲ ✲ ✲ Re{ω} ✲ Re{ω} ✲ Re{ω} ✲ Re{ω} ✲ Re{ω} ✲ Re{ω} ✲ Re{ω} 27 0 ω-plane (b) g R ✿ ✲ 0 ω-plane (c) g A ③ 0 (d) g − ✿ ✲ ✲ ω-plane ✲ 0 ω-plane (e) g (f) g < ③ ✛ ✲ > ✿ 0 ω-plane 0 ω-plane (g) g ✿ ③ ✲ ✛ ✛ ✲ 0 Fig. 2. Integration paths (solid lines) in the ω-plane for obtaining various Green’s functions satisfying a second-order (in time) differential equation. The singularities g are differences of the integrand are located on the real ω-axis.

9. Show that the limit L → ∞ of the integral of the local DOS R(r, E) = E −∞ dE (r, E ), develops finite discontinuities (most of them exponentially 52 3 Physical Significance of G. Application to the Free-Particle Case small) for almost all energies E. Hence R(E) cannot be defined in the limit L → ∞ since it develops singularities almost everywhere of the form known as the devil’s staircase. On the other hand, if the states are ordinary propagating states (extended over the whole “volume” Ld ), the number of states R(E) approaches a regular function as L → ∞.

2). We have excluded the solution corresponding to g − (τ ) on the basis of the physical argument that the response of a system at a time t depends only on what the source, f (r , t ), was in the past (t < t). 21) as t ψ (r, t) = φ(r, t) + dr −∞ dt g (r, r , t − t ) f (r , t ) . 1 Examples Here we calculate the various Green’s functions for the case L = −∇2 . 10). 13); φn (r) = eik · r / Ω, and λn = k 2 . 1 First-Order Case 25 ≡ r − r and d is the dimensionality. 24) where π/icτ is the square root with a positive real part.

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