Quantum Gravity 319.6: More on Green's Function of 319.5
- From: OsherD <mdoctorow@xxxxxxxxx>
- Date: Sat, 11 Apr 2009 20:57:08 -0700 (PDT)
From Osher Doctorow
See "Green's function" and "Fredholm Theory" and "Fredholm integral"
in Wikipedia for example, or in Wolfram's site (Eric Weisstein, etc.).
A Green's function can be thought of as a right inverse of a linear
operator L, even though not every such operator has a Green's
function.
It can also be thought of as an expansion of a function f in a Dirac
Delta function basis (f is projcted over delta(x - s)) and then the
solutions are superposed on each projection. This yields an integral,
the Fredholm integral, and an integral equation.
Fredholm Theory is concerned with solutions of integral equations of
form:
1) g(x) = I[K(x, y)f(y)]dy, I...dy integral from a to h
which arise naturally in many problems in physics as inverses of
differential equations of form:
2) L(g(x)) = f(x), g unknown, f given, L linear differential operator
The general method is to try to solve:
3) L(K(x, y)) = delta(x - y)
and then the solution to the differential equation (2) is written:
4) g(x) = I[K(x, y)f(y)]dy
where K(x, y) is called the Green's fuction.
One can also solve the inhomogeneous Fredholm integral equation:
5) f(x) = -w phi(x) + I[K(x, y)phi(y)]dy
formally written:
6) f = (K - w) phi
with solution:
7) phi = [1/(K - w)]f
and solutions of this form are called the Resolvent Formalism. The
Resolvent is defined as the operator:
8) R(w) = [1/(K - wI)]
or in alternate form:
9) R(lambda) = 1/[I - lambda K]
The Fredholm determinant is:
10) Fredholm determinant = det(I - lambda K)
The zeta function, which is the same general type as the Riemann zeta
function discussed earlier except that K is not known, is:
11) zeta(s) = 1/det(I - sK)
and can be interpreted as the determinant of the resolvent and is
important in studying dynamical systems.
Osher Doctorow
.
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