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The general study of Green's function written in the above form, and its relationship to the function spaces formed by the eigenvectors, is known as Fredholm theory. There are several other methods for finding Green's functions, including the method of images, separation of variables, and Laplace transforms. [1]
is the derivative of the Green's function along the inward-pointing unit normal vector ^. The integration is performed on the boundary, with measure d s {\displaystyle ds} . The function ν ( s ) {\displaystyle \nu (s)} is given by the unique solution to the Fredholm integral equation of the second kind,
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...
The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point "Green's functions" in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the ...
A nonlinear operator = (,, | |) is elliptic if its linearization is; i.e. the first-order Taylor expansion with respect to u and its derivatives about any point is an elliptic operator. Example 1 The negative of the Laplacian in R d given by − Δ u = − ∑ i = 1 d ∂ i 2 u {\displaystyle -\Delta u=-\sum _{i=1}^{d}\partial _{i}^{2}u} is a ...
A Green's function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. Other methods for obtaining Green's functions include the method of images, separation of variables, and Laplace transforms (Cole, 2011).
The name "Green measure" comes from the fact that if X is Brownian motion, then (,) = (,), where G(x, y) is Green's function for the operator L X (which, in the case of Brownian motion, is 1 / 2 Δ, where Δ is the Laplace operator) on the domain D.
In many-body physics, the problem of analytic continuation is that of numerically extracting the spectral density of a Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating dynamical properties of physical systems from Quantum Monte Carlo simulations, which often compute Green function ...