Editing Spectral theory (section)

==Resolution of the identity==
{{See also |Borel functional calculus#Resolution of the identity}}

This section continues in the rough and ready manner of the above section using the bra–ket notation, and glossing over the many important details of a rigorous treatment.<ref name="Vujičić">

See discussion in Dirac's book referred to above, and {{Cite book |title=Linear algebra thoroughly explained |author=Milan Vujičić |url= https://books.google.com/books?id=pifStNLaXGkC&pg=PA274 |page=274 |isbn=978-3-540-74637-9 |year=2008 |publisher=Springer }}</ref> A rigorous mathematical treatment may be found in various references.<ref name=rigor>See, for example, the fundamental text of {{Cite book |title=''op. cit'' |author=John von Neumann |year=1955 |publisher=Princeton University Press |url=https://books.google.com/books?id=JLyCo3RO4qUC|isbn=0-691-02893-1 }} and {{Cite book
|title=Linear Operator Theory in Engineering and Science; ''Vol. 40 of'' Applied mathematical science  |page=401 |url=https://books.google.com/books?id=t3SXs4-KrE0C&pg=PA401 |author=Arch W. Naylor, George R. Sell |isbn=0-387-95001-X |publisher=Springer |year=2000}}, {{Cite book
|title=Advanced linear algebra |author=Steven Roman |url=https://books.google.com/books?id=bSyQr-wUys8C&pg=PA233 |isbn=978-0-387-72828-5 |edition=3rd |year=2008 |publisher=Springer}}, {{Cite book |title=Expansions in eigenfunctions of selfadjoint operators; ''Vol. 17 in'' Translations of mathematical monographs |url=https://books.google.com/books?id=OPPWBE3WQqkC&pg=PA317 |author=I︠U︡riĭ Makarovich Berezanskiĭ |isbn=0-8218-1567-9 |year=1968 |publisher=American Mathematical Society}}</ref> In particular, the dimension ''n'' of the space will be finite.

Using the bra–ket notation of the above section, the identity operator may be written as:

:<math>I = \sum _{i=1} ^{n} | e_i \rangle \langle f_i |  </math>

where it is supposed as above that <math>\{ |e_i\rangle\}</math> are a [[Basis (linear algebra)|basis]] and the <math> \{ \langle f_i | \}</math> a reciprocal basis for the space satisfying the relation:

:<math>\langle f_i | e_j\rangle = \delta_{ij} . </math>

This expression of the identity operation is called a ''representation'' or a ''resolution'' of the identity.<ref name= "Vujičić"/><ref name= rigor/> This formal representation satisfies the basic property of the identity:
:<math> I^k = I </math>
valid for every positive integer ''k''.

Applying the resolution of the identity to any function in the space <math>| \psi \rangle</math>, one obtains:

:<math>I |\psi \rangle = |\psi \rangle = \sum_{i=1}^{n} | e_i \rangle \langle f_i | \psi \rangle = \sum_{i=1}^{n} c_i | e_i \rangle</math>
which is the [[Generalized_Fourier_series|generalized Fourier expansion]] of ψ in terms of the basis functions {&nbsp;e<sub>i</sub>&nbsp;}.<ref name=Folland>
See for example, {{cite book |author=Gerald B Folland |title=Fourier Analysis and its Applications |publisher=American Mathematical Society |edition=Reprint of Wadsworth & Brooks/Cole 1992 |chapter-url=https://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |pages = 77 ''ff'' |chapter=Convergence and completeness |year=2009 |isbn=978-0-8218-4790-9}}
</ref>
Here <math>c_i = \langle f_i | \psi \rangle</math>.

Given some operator equation of the form:
:<math>O | \psi \rangle = | h \rangle  </math>
with ''h'' in the space, this equation can be solved in the above basis through the formal manipulations:
:<math> O | \psi \rangle = \sum_{i=1}^{n} c_i \left( O | e_i \rangle \right)  =  \sum_{i=1}^{n} | e_i \rangle \langle f_i |  h \rangle , </math>
:<math>\langle f_j|O| \psi \rangle = \sum_{i=1}^{n}  c_i \langle f_j| O | e_i \rangle  =  \sum_{i=1}^{n} \langle f_j| e_i \rangle \langle f_i | h \rangle  = \langle f_j |  h \rangle, \quad \forall j </math>
which converts the operator equation to a [[matrix equation]] determining the unknown coefficients ''c<sub>j</sub>'' in terms of the generalized Fourier coefficients <math>\langle f_j | h \rangle</math> of ''h'' and the matrix elements <math>O_{ji}= \langle f_j| O | e_i \rangle </math> of the operator ''O''.

The role of spectral theory arises in establishing the nature and existence of the basis and the reciprocal basis. In particular, the basis might consist of the eigenfunctions of some linear operator ''L'':

:<math>L | e_i \rangle = \lambda_i | e_i \rangle \, ; </math>

with the {&nbsp;''λ<sub>i</sub>''&nbsp;} the eigenvalues of ''L'' from the spectrum of ''L''. Then the resolution of the identity above provides the dyad expansion of ''L'':

:<math>LI = L = \sum_{i=1}^{n} L | e_i \rangle \langle f_i|  = \sum_{i=1}^{n} \lambda _i | e_i \rangle \langle f_i | . </math>