Editing Spectral theory (section)

==Operator equations==
{{See also|Spectral theory of ordinary differential equations|Integral equation}}
Consider the operator equation:

:<math>(O-\lambda I ) |\psi \rangle = |h \rangle; </math>

in terms of coordinates:

:<math>\int \langle x, (O-\lambda I)y \rangle \langle y, \psi \rangle \, dy = h(x). </math>

A particular case is ''λ'' = 0.

The Green's function of the previous section is:

:<math>\langle y, G(\lambda) z\rangle = \left \langle y, (O-\lambda I)^{-1} z \right \rangle = G(y, z; \lambda),</math>

and satisfies:

:<math>\int \langle x, (O - \lambda I) y \rangle \langle y, G(\lambda) z \rangle \, dy = \int \langle x, (O-\lambda I) y \rangle \left \langle y, (O-\lambda I)^{-1} z \right \rangle \, dy = \langle x , z \rangle = \delta (x-z).</math>

Using this Green's function property:

:<math>\int \langle x, (O-\lambda I) y \rangle G(y, z; \lambda ) \, dy = \delta (x-z). </math>

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

:<math>\int dz \, h(z) \int dy \, \langle x, (O-\lambda I)y \rangle G(y, z; \lambda)=\int dy \, \langle x, (O-\lambda I) y \rangle \int dz \, h(z)G(y, z; \lambda) = h(x), </math>

which suggests the solution is:

:<math>\psi(x) = \int h(z) G(x, z; \lambda) \, dz.</math>

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:

:<math>G(x, z; \lambda)  = \sum_{i=1}^n \frac{e_i (x) f_i^*(z)}{\lambda - \lambda_i}.</math>

There are many other ways to find ''G'', of course.<ref name=Green>

For example, see {{cite book |title=Mathematical physics: a modern introduction to its foundations |author= Sadri Hassani |chapter=Chapter 20: Green's functions in one dimension |page=553 ''et seq'' |publisher=Springer |chapter-url=https://books.google.com/books?id=BCMLOp6DyFIC&pg=RA1-PA553 |year=1999 |isbn=0-387-98579-4}} and {{cite book |title=Green's function and boundary elements of multifield materials |author=Qing-Hua Qin |url=https://books.google.com/books?id=UUfy8CcJiDkC|isbn=978-0-08-045134-3 |year=2007 |publisher=Elsevier}}</ref> See the articles on [[Green's function#Green.27s functions for solving inhomogeneous boundary value problems|Green's functions]] and on [[Fredholm theory#Inhomogeneous equations|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]], [[Distribution (mathematics)|distributions]] and so forth. Consult these articles and the references for more detail.