Editing Spectral theory (section)

==Resolvent operator==
{{Main|Resolvent formalism}}
{{See also|Green's function|Dirac delta function}}
Using spectral theory, the resolvent operator ''R'':

:<math>R =  (\lambda I - L)^{-1},\, </math>

can be evaluated  in terms of the eigenfunctions and eigenvalues of ''L'', and the Green's function corresponding to ''L'' can be found.

Applying ''R'' to some arbitrary function in the space, say <math>\varphi</math>,

:<math>R  |\varphi \rangle = (\lambda I - L)^{-1} |\varphi \rangle = \sum_{i=1}^n \frac{1}{\lambda- \lambda_i} |e_i \rangle \langle f_i | \varphi \rangle. </math>

This function has [[Pole (complex analysis)|poles]] in the complex ''λ''-plane at each eigenvalue of ''L''. Thus, using the [[calculus of residues]]:

:<math>\frac{1}{2\pi i } \oint_C R  |\varphi \rangle d \lambda = -\sum_{i=1}^n |e_i \rangle   \langle f_i | \varphi \rangle  = -|\varphi \rangle,</math>

where the [[line integral]] is over a contour ''C'' that includes all the eigenvalues of ''L''.

Suppose our functions are defined over some coordinates {''x<sub>j</sub>''}, that is:

:<math>\langle x, \varphi \rangle = \varphi (x_1, x_2, ...). </math>

Introducing the notation

:<math> \langle x , y \rangle = \delta (x-y), </math>

where ''δ(x − y)'' = ''δ(x<sub>1</sub> − y<sub>1</sub>, x<sub>2</sub> − y<sub>2</sub>, x<sub>3</sub> − y<sub>3</sub>,  ...)'' is the [[Dirac delta function]],<ref name=Dirac3>{{cite book |url=https://books.google.com/books?id=XehUpGiM6FIC&pg=PA60 |page=60 ''ff'' |author=PAM Dirac |title=''op. cit'' |year=1981 |publisher=Clarendon Press |isbn=0-19-852011-5}}</ref>
we can write

:<math>\langle x, \varphi \rangle = \int \langle x , y \rangle \langle y, \varphi \rangle dy. </math>

Then:

:<math>\begin{align}
\left\langle x, \frac{1}{2\pi i } \oint_C \frac{\varphi}{\lambda I - L} d \lambda\right\rangle &= \frac{1}{2\pi i }\oint_C d \lambda \left \langle x, \frac{\varphi}{\lambda I - L} \right \rangle\\
&= \frac{1}{2\pi i } \oint_C d \lambda \int dy \left \langle x,  \frac{y}{\lambda I - L} \right \rangle  \langle y, \varphi \rangle
\end{align}</math>

The function ''G(x, y; λ)'' defined by:

:<math>\begin{align}
G(x, y; \lambda) &= \left \langle x, \frac{y}{\lambda I - L} \right \rangle \\
&= \sum_{i=1}^n \sum_{j=1}^n \langle x, e_i \rangle \left \langle f_i, \frac{e_j}{\lambda I - L} \right \rangle \langle f_j , y\rangle \\
&= \sum_{i=1}^n \frac{\langle x,  e_i \rangle \langle f_i , y\rangle }{\lambda  - \lambda_i} \\
&= \sum_{i=1}^n \frac{e_i (x) f_i^*(y) }{\lambda  - \lambda_i},
\end{align}</math>

is called the ''[[Green's function]]'' for operator ''L'', and satisfies:<ref name=Friedman3>{{cite book |title=''op. cit'' |page=214, Eq. 2.14 |author=Bernard Friedman |year=1956 |publisher=Dover Publications |isbn=0-486-66444-9 |url=https://books.google.com/books?id=gnQeAQAAIAAJ&q=intitle:applied+intitle:mathematics+inauthor:Friedman }}</ref>

:<math>\frac{1}{2\pi i }\oint_C G(x,y;\lambda) \, d \lambda = -\sum_{i=1}^n  \langle x, e_i \rangle \langle f_i , y\rangle = -\langle x, y\rangle = -\delta (x-y). </math>