Mathematical Methods of Physics/General theory

A Green's function for a linear operator over is a real function such that is solved by ; where symbolizes convolution. Hence, so that is a right inverse of and is a particular solution to the inhomogeneous equation.

For example: is solved by

Such a function might not exist and when it does might not be unique. The conditions under which this method is valid require careful examination. However, the theory of Green's functions obtains a more complete and regular form over the theory of distributions, or generalized functions.

As will be seen, the theory of Green's functions provides an extremely elegant procedure of solving differential equations. We wish to present here this method on a rigorous foundation.

The Dirac delta-functionEdit

The Dirac delta-function  is not a function as it is ordinarily defined. However, we write it as if it were a function, keeping in mind the scope of the definition.

For any function  ,we define

  but for every  ,


It follows that  

These conditions seem to be satisfied by a "function"   which has value zero whenever  , but has "infinite" value at  


There are a few ways to approximate the delta function in terms of sequences ordinary functions. We give two examples

The Boxcar functionEdit

The boxcar function   such that


We can see that the sequence   represents an approximation to the delta function.

The bell curveEdit

The delta function can also be approximated by the ubiquitous Gaussian.

We write  

Green's functionEdit

Consider an equation of the type  ...(1), where   is a differential operator. The functions   may in general be functions of several independents, but for sake of clarity, we will write them here as if they were real valued. In most cases of interest, this equation can be written in the form

  to be solved for   in some closed set  , with   being non-zero over  

Now, it so happens, that in problems of physics, it is much more convenient to solve the equation  , when   is the delta function  .

In this case, the solution of the operator   is called the Green's function  . That is,


Now, by the definition of the delta-function, we have that  , where   act as "weights" to the delta function.

Hence, we have,  

Note here that  is an operator that depends on   but not  . Thus,

 . We can view this as analogous to the inversion of   and hence, we write


The subscript   denotes that we have found a particular solution among the many possible. For example, consider any harmonic solution  .

If we add  , we see that   is still a solution of (1). Thus, we have a class of functions satisfying (1).

Boundary value problemsEdit

Problems of physics are often presented as the operator equation   to be solved for   on a closed set  , together with the boundary condition that   for all   (  is the boundary of  ).

  is a given function satisfying   that describes the behaviour of the solution at the boundary of the region of concern.

Thus if a problem is stated as



to be solved for   over a closed set  ,

The solution can be given as  

Green's functions from eigenfunctionsEdit

Consider the eigenvalues   and the corresponding eigenfunctions   of the differential operator  , that is  

Without loss of generality, we assume that these eigenfunctions are orthogonal. Further, we assume that they form a basis.

Thus, we can write   and  .

Now   and hence,  

by definition of orthogonality,  


and hence, we can write the Green's function as