Spectral Theory - Operator Equations

Operator Equations

See also: Spectral theory of ordinary differential equations and Integral equation

Consider the operator equation:

in terms of coordinates:

A particular case is λ = 0.

The Green's function of the previous section is:

and satisfies:

  

Using this Green's function property:

Then, multiplying both sides of this equation by h(z) and integrating:

 

which suggests the solution is:

That is, the function ψ(x) satisfying the operator equation is found if we can find the spectrum of O, and construct G, for example by using:

There are many other ways to find G, of course. See the articles on Green's functions and on Fredholm integral equations. It must be kept in mind that the above mathematics is purely formal, and a rigorous treatment involves some pretty sophisticated mathematics, including a good background knowledge of functional analysis, Hilbert spaces, distributions and so forth. Consult these articles and the references for more detail.