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| In [[mathematics]], '''spectral theory''' is an inclusive term for theories extending the [[eigenvector]] and [[eigenvalue]] theory of a single [[square matrix]] to a much broader theory of the structure of [[operator (mathematics)|operator]]s in a variety of [[mathematical space]]s.<ref name=Dieudonné>{{cite book |title=History of functional analysis |author=Jean Alexandre Dieudonné |url=http://books.google.com/books?id=mg7r4acKgq0C&printsec=frontcover |isbn=0-444-86148-3 |year=1981 |publisher=Elsevier}}</ref> It is a result of studies of [[linear algebra]] and the solutions of [[System of linear equations|systems of linear equations]] and their generalizations.<ref name=Arveson> {{cite book |title=A short course on spectral theory |author=William Arveson |chapter=Chapter 1: spectral theory and Banach algebras |url =http://books.google.com/books?id=ARdehHGWV1QC |isbn=0-387-95300-0 |year=2002 |publisher=Springer}}</ref> The theory is connected to that of [[analytic functions]] because the spectral properties of an operator are related to analytic functions of the spectral parameter.<ref name="Sadovnichiĭ">{{cite book |title=Theory of Operators |author=Viktor Antonovich Sadovnichiĭ |chapter=Chapter 4: The geometry of Hilbert space: the spectral theory of operators |url=http://books.google.com/books?id=SR1QkG6OkVEC&pg=PA181&lpg=PA181 |page= 181 ''et seq'' |isbn=0-306-11028-8 |year=1991 |publisher=Springer}}
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| </ref>
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| ==Mathematical background==
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| The name ''spectral theory'' was introduced by [[David Hilbert]] in his original formulation of [[Hilbert space]] theory, which was cast in terms of [[quadratic form]]s in infinitely many variables. The original [[spectral theorem]] was therefore conceived as a version of the theorem on [[principal axes]] of an [[ellipsoid]], in an infinite-dimensional setting. The later discovery in [[quantum mechanics]] that spectral theory could explain features of [[Emission spectrum|atomic spectra]] was therefore fortuitous.
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| There have been three main ways to formulate spectral theory, all of which retain their usefulness. After Hilbert's initial formulation, the later development of abstract [[Hilbert space]] and the spectral theory of a single [[normal operator]] on it did very much go in parallel with the requirements of [[physics]]; particularly at the hands of [[von Neumann]].<ref name= vonNeumann>
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| {{Cite book |title=The mathematical foundations of quantum mechanics; ''Volume 2 in Princeton'' Landmarks in Mathematics ''series'' |author=John von Neumann |url=http://books.google.com/books?id=JLyCo3RO4qUC&printsec=frontcover&dq=mathematical+foundations+of+quantum+mechanics+inauthor:von+inauthor:neumann&cd=1#v=onepage&q=&f=false |isbn=0-691-02893-1 |year=1996 |publisher =Princeton University Press |edition=Reprint of translation of original 1932 }}
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| </ref> The further theory built on this to include [[Banach algebra]]s, which can be given abstractly. This development leads to the [[Gelfand representation]], which covers the [[commutative Banach algebra|commutative case]], and further into [[non-commutative harmonic analysis]].
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| The difference can be seen in making the connection with [[Fourier analysis]]. The [[Fourier transform]] on the [[real line]] is in one sense the spectral theory of [[derivative|differentiation]] ''qua'' [[differential operator]]. But for that to cover the phenomena one has already to deal with [[generalized eigenfunction]]s (for example, by means of a [[rigged Hilbert space]]). On the other hand it is simple to construct a [[group algebra]], the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of [[Pontryagin duality]].
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| One can also study the spectral properties of operators on [[Banach spaces]]. For example, [[compact operator]]s on Banach spaces have many spectral properties similar to that of [[Matrix (mathematics)|matrices]].
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| ==Physical background==
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| The background in the physics of [[vibration]]s has been explained in this way:<ref name=Davies>[[E. Brian Davies]], quoted on the King's College London analysis group website {{Cite web |url=http://www.kcl.ac.uk/schools/pse/maths/research/analysis/research.html |title=Research at the analysis group}}</ref>
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| {{Cquote|Spectral theory is connected with the investigation of localized vibrations of a variety of different objects, from [[atom]]s and [[molecule]]s in [[chemistry]] to obstacles in [[Waveguide (acoustics)|acoustic waveguide]]s. These vibrations have [[frequency|frequencies]], and the issue is to decide when such localized vibrations occur, and how to go about computing the frequencies. This is a very complicated problem since every object has not only a [[fundamental tone]] but also a complicated series of [[overtone]]s, which vary radically from one body to another.}}
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| The mathematical theory is not dependent on such physical ideas on a technical level, but there are examples of mutual influence (see for example [[Mark Kac]]'s question ''[[Can you hear the shape of a drum?]]''). Hilbert's adoption of the term "spectrum" has been attributed to an 1897 paper of [[Wilhelm Wirtinger]] on [[Hill differential equation]] (by [[Jean Dieudonné]]), and it was taken up by his students during the first decade of the twentieth century, among them [[Erhard Schmidt]] and [[Hermann Weyl]]. The conceptual basis for [[Hilbert space]] was developed from Hilbert's ideas by [[Erhard Schmidt]] and [[Frigyes Riesz]].<ref name=Young>{{Cite book |title=An introduction to Hilbert space |author=Nicholas Young |url=http://books.google.com/books?id=_igwFHKwcyYC&pg=PA3 |page=3 |isbn=0-521-33717-8 |publisher=Cambridge University Press |year=1988}}</ref><ref name=Dorier>{{Cite book |title=On the teaching of linear algebra; ''Vol. 23 of'' Mathematics education library |author=Jean-Luc Dorier |url=http://books.google.com/books?id=gqZUGMKtNuoC&pg=PA50&dq=%22thinking+geometrically+in+Hilbert%27s+%22&cd=1#v=onepage&q=%22thinking%20geometrically%20in%20Hilbert%27s%20%22&f=false |isbn=0-7923-6539-9 |publisher=Springer |year=2000 }}
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| </ref> It was almost twenty years later, when [[quantum mechanics]] was formulated in terms of the [[Schrödinger equation]], that the connection was made to [[atomic spectra]]; a connection with the mathematical physics of vibration had been suspected before, as remarked by [[Henri Poincaré]], but rejected for simple quantitative reasons, absent an explanation of the [[Balmer series]].<ref>Cf. [http://www.dm.unito.it/personalpages/capietto/Spectra.pdf Spectra in mathematics and in physics] by Jean Mawhin, p.4 and pp. 10-11.</ref> The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous, rather than being an object of Hilbert's spectral theory.
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| ==A definition of spectrum==
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| {{main|Spectrum (functional analysis)}}
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| Consider a [[Bounded linear operator|bounded linear transformation]] ''T'' defined everywhere over a general [[Banach space]]. We form the transformation:
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| :<math> R_{\zeta} = \left( \zeta I - T \right)^{-1}.</math>
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| Here ''I'' is the [[identity operator]] and ζ is a [[complex number]]. The ''inverse'' of an operator ''T'', that is ''T''<sup>−1</sup>, is defined by:
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| :<math>T T^{-1} = T^{-1} T = I. </math>
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| If the inverse exists, ''T'' is called ''regular''. If it does not exist, ''T'' is called ''singular''.
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| With these definitions, the ''[[resolvent set]]'' of ''T'' is the set of all complex numbers ζ such that ''R<sub>ζ</sub>'' exists and is [[Bounded operator|bounded]]. This set often is denoted as ''ρ(T)''. The ''spectrum'' of ''T'' is the set of all complex numbers ζ such that ''R<sub>ζ</sub>'' <u>fails</u> to exist or is unbounded. Often the spectrum of ''T'' is denoted by ''σ(T)''. The function ''R<sub>ζ</sub>'' for all ζ in ''ρ(T)'' (that is, wherever ''R<sub>ζ</sub>'' exists as a bounded operator) is called the [[Resolvent formalism|resolvent]] of ''T''. The ''spectrum'' of ''T'' is therefore the complement of the ''resolvent set'' of ''T'' in the complex plane.<ref name=Lorch>
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| {{Cite book |title=Spectral Theory |author=Edgar Raymond Lorch |year=2003 |edition=Reprint of Oxford 1962 |page=89 |publisher=Textbook Publishers |isbn=0-7581-7156-0 |url=http://books.google.com/books?id=X3U2AAAACAAJ&dq=intitle:spectral+intitle:theory+inauthor:Lorch&cd=1}}</ref> Every [[eigenvalue]] of ''T'' belongs to ''σ(T)'', but ''σ(T)'' may contain non-eigenvalues.<ref name= Young2>{{cite book |title=''op. cit'' |author= Nicholas Young |url=http://books.google.com/books?id=_igwFHKwcyYC&pg=PA81 |page=81 |isbn=0-521-33717-8}}</ref>
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| This definition applies to a Banach space, but of course other types of space exist as well, for example, [[topological vector spaces]] include Banach spaces, but can be more general.<ref name=Wolff>{{Cite book |title=Topological vector spaces |author=Helmut H. Schaefer, Manfred P. H. Wolff |url=http://books.google.com/books?id=9kXY742pABoC&pg=PA36 |page=36 |year=1999 |isbn=0-387-98726-6 |edition=2nd |publisher=Springer}}</ref><ref name= Zhelobenko>
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| {{Cite book |title=Principal structures and methods of representation theory |author=Dmitriĭ Petrovich Zhelobenko |url=http://books.google.com/books?id=3TkmvZktjp8C&pg=PA24 |isbn= 0821837311 |publisher=American Mathematical Society |year=2006}}</ref> On the other hand, Banach spaces include [[Hilbert space]]s, and it is these spaces that find the greatest application and the richest theoretical results.<ref name=Lorch2>{{Cite book |title=''op. cit.'' |author=Edgar Raymond Lorch |year=2003 |isbn=0-7581-7156-0 |url=http://books.google.com/books?id=X3U2AAAACAAJ&dq=intitle:spectral+intitle:theory+inauthor:Lorch&cd=1 |page=57 |chapter=Chapter III: Hilbert Space}} </ref> With suitable restrictions, much can be said about the structure of the [[Hilbert_space#Spectral_theory|spectra of transformations]] in a Hilbert space. In particular, for [[self-adjoint operator]]s, the spectrum lies on the [[real line]] and (in general) is a [[Decomposition of spectrum (functional analysis)|spectral combination]] of a point spectrum of discrete [[Eigenvalues#Computation_of_eigenvalues.2C_and_the_characteristic_equation|eigenvalues]] and a [[continuous spectrum]].<ref name=Lorch3>{{Cite book |title=''op. cit.'' |author=Edgar Raymond Lorch |year=2003 |isbn=0-7581-7156-0 |url=http://books.google.com/books?id=X3U2AAAACAAJ&dq=intitle:spectral+intitle:theory+inauthor:Lorch&cd=1 |page=106 ''ff'' |chapter=Chapter V: The Structure of Self-Adjoint Transformations}} </ref>
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| ==Spectral theory briefly==
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| {{Main|Spectral theorem}}
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| {{See also|Eigenvalue, eigenvector and eigenspace}}
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| In [[functional analysis]] and [[linear algebra]] the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators. As a full rigorous presentation is not appropriate for this article, we take an approach that avoids much of the rigor and satisfaction of a formal treatment with the aim of being more comprehensible to a non-specialist.
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| This topic is easiest to describe by introducing the [[bra-ket notation]] of [[Paul Dirac|Dirac]] for operators.<ref name= Friedman>{{Cite book |title=Principles and Techniques of Applied Mathematics |author=Bernard Friedman |year=1990 |publisher=Dover Publications |page=26 |isbn=0-486-66444-9 |url=http://books.google.com/books?id=gnQeAQAAIAAJ&dq=intitle:applied+intitle:mathematics+inauthor:Friedman&cd=1 |edition=Reprint of 1956 Wiley}} </ref><ref name=Dirac>{{Cite book |title=The principles of quantum mechanics |author=PAM Dirac |edition=4th |isbn=0-19-852011-5 |publisher=Oxford University Press |year=1981 |page=29 ''ff'' |url=http://books.google.com/books?id=XehUpGiM6FIC&pg=PA29}}</ref> As an example, a very particular linear operator ''L'' might be written as a [[dyadic product]]:<ref name=Audretsch>{{Cite book |title=Entangled systems: new directions in quantum physics |author=Jürgen Audretsch |page=5 |url=http://books.google.com/books?id=8NxIgwAOU6IC&pg=PA5 |chapter=Chapter 1.1.2: Linear operators on the Hilbert space |isbn=3-527-40684-0 |publisher=Wiley-VCH |year=2007}}</ref><ref name=Howland>{{Cite book |title=Intermediate dynamics: a linear algebraic approach |url=http://books.google.com/books?id=SepP8-W3M0AC&pg=PA69&dq=dyad+representation+operator&cd=32#v=onepage&q=dyad%20representation%20operator&f=false |page=69 ''ff'' |author=R. A. Howland |publisher=Birkhäuser |year=2006 |isbn=0-387-28059-6 |edition=2nd}}</ref>
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| :<math> L = | k_1 \rangle \langle b_1 |, </math>
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| in terms of the "bra" ''<math> \langle b_1 | </math>'' and the "ket" ''<math> | k_1 \rangle </math>'' . A function ''f'' is described by a ''ket'' as ''<math> | f \rangle </math>''. The function ''f''(''x'') defined on the coordinates <math>(x_1, x_2, x_3, \dots)</math> is denoted as:
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| :<math> f(x)=\langle x, f\rangle </math>
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| and the magnitude of ''f'' by:
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| :<math> \|f \|^2 = \langle f, f\rangle =\int \langle f, x\rangle \langle x, f \rangle \, dx = \int f^*(x) f(x) \, dx </math>
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| where the notation '*' denotes a [[complex conjugate]]. This [[inner product]] choice defines a very specific [[inner product space]], restricting the generality of the arguments that follow.<ref name=Lorch2/>
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| The effect of ''L'' upon a function ''f'' is then described as:
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| :<math> L | f\rangle = | k_1 \rangle \langle b_1 | f \rangle </math>
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| expressing the result that the effect of ''L'' on ''f'' is to produce a new function <math> | k_1 \rangle </math> multiplied by the inner product represented by <math>\langle b_1 | f \rangle </math>.
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|
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| A more general linear operator ''L'' might be expressed as:
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| :<math> L = \lambda_1 | e_1\rangle\langle f_1| + \lambda_2 | e_2\rangle \langle f_2| + \lambda_3 | e_3\rangle\langle f_3| + \dots , </math>
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| where the <math> \{ \, \lambda_i \, \}</math> are scalars and the <math> \{ \, | e_i \rangle \, \} </math> are a [[Basis (linear algebra)|basis]] and the <math> \{ \, \langle f_i | \, \} </math> a [[Dual basis|reciprocal basis]] for the space. The relation between the basis and the reciprocal basis is described, in part, by:
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| :<math> \langle f_i | e_j \rangle = \delta_{ij} </math> | |
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| If such a formalism applies, the <math> \{ \, \lambda_i \, \}</math> are [[eigenvalues]] of ''L'' and the functions <math> \{ \, | e_i \rangle \, \} </math> are [[eigenfunctions]] of ''L''. The eigenvalues are in the ''spectrum'' of ''L''.<ref name= Friedman2>{{Cite book |title=op. cit. |author=Bernard Friedman |year=1990 |page=57 |chapter=Chapter 2: Spectral theory of operators |isbn=0-486-66444-9 |url=http://books.google.com/books?id=gnQeAQAAIAAJ&dq=intitle:applied+intitle:mathematics+inauthor:Friedman&cd=1}} </ref>
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| Some natural questions are: under what circumstances does this formalism work, and for what operators ''L'' are expansions in series of other operators like this possible? Can any function ''f'' be expressed in terms of the eigenfunctions (are they a [[Schauder basis]]) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite dimensional spaces and finite dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in [[functional analysis]] and [[Matrix (mathematics)|matrix algebra]].
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| ==Resolution of the identity==
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| {{See also |Borel functional calculus#Resolution of the identity}}
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| 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">
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| See discussion in Dirac's book referred to above, and {{Cite book |title=Linear algebra thoroughly explained |author=Milan Vujičić |url= http://books.google.com/books?id=pifStNLaXGkC&pg=PA274 |page=274 |isbn=3-540-74637-4 |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 |url=http://books.google.com/books?id=JLyCo3RO4qUC&printsec=frontcover|isbn=0-691-02893-1 }} and {{Cite book
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| |title=Linear Operator Theory in Engineering and Science; ''Vol. 40 of'' Applied mathematical science |page=401 |url=http://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
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| |title=Advanced linear algebra |author=Steven Roman |url=http://books.google.com/books?id=bSyQr-wUys8C&pg=PA233 |isbn=0-387-72828-7 |edition=3rd |year=2008 |publisher=Springer}}, {{Cite book |title=Expansions in eigenfunctions of selfadjoint operators; ''Vol. 17 in'' Translations of mathematical monographs |url=http://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.
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| Using the bra-ket notation of the above section, the identity operator may be written as:
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| :<math>I = \sum _{i=1} ^{n} | e_i \rangle \langle f_i | </math>
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| 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:
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| :<math>\langle f_i | e_j\rangle = \delta_{ij} . </math> | |
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| This expression of the identity operation is called a ''representation'' or a ''resolution'' of the identity.<ref name= "Vujičić"/><sup>,</sup><ref name= rigor/> This formal representation satisfies the basic property of the identity:
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| :<math> I^k = I\, </math>
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| valid for every positive integer ''k''.
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| Applying the resolution of the identity to any function in the space ''<math>| \psi \rangle</math>'', one obtains:
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| :<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>
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| which is the generalized [[Fourier series|Fourier expansion]] of ψ in terms of the basis functions { e<sub>i</sub> }.<ref name=Folland>
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| 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 |url=http://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |pages = 77 ''ff'' |chapter=Convergence and completeness |year=2009 |isbn=0-8218-4790-2}}
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| </ref>
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| Here ''<math>c_i = \langle f_i | \psi \rangle</math>''.
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| Given some operator equation of the form:
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| :<math>O | \psi \rangle = | h \rangle </math>
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| with ''h'' in the space, this equation can be solved in the above basis through the formal manipulations:
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| :<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>
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| :<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>
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| 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''.
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| 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'':
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| :<math>L | e_i \rangle = \lambda_i | e_i \rangle \, ; </math>
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| with the { ''λ<sub>i</sub>'' } the eigenvalues of ''L'' from the spectrum of ''L''. Then the resolution of the identity above provides the dyad expansion of ''L'':
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| :<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>
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| ==Resolvent operator==
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| {{main|Resolvent formalism}}
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| {{See also|Green's function|Dirac delta function}}
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| Using spectral theory, the resolvent operator ''R'':
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| :<math>R = (\lambda I - L)^{-1},\, </math>
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| can be evaluated in terms of the eigenfunctions and eigenvalues of ''L'', and the Green's function corresponding to ''L'' can be found.
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| Applying ''R'' to some arbitrary function in the space, say <math>\varphi</math>,
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| :<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>
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| This function has [[Pole (complex analysis)|poles]] in the complex ''λ''-plane at each eigenvalue of ''L''. Thus, using the [[calculus of residues]]:
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| :<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>
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| where the [[line integral]] is over a contour ''C'' that includes all the eigenvalues of ''L''.
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| Suppose our functions are defined over some coordinates {''x<sub>j</sub>''}, that is:
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| :<math>\langle x, \varphi \rangle = \varphi (x_1, x_2, ...). </math>
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| Introducing the notation
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| :<math> \langle x , y \rangle = \delta (x-y), </math>
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| 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=http://books.google.com/books?id=XehUpGiM6FIC&pg=PA60#v=onepage&q=&f=false |page=60 ''ff'' |author=PAM Dirac |title=''op. cit'' |isbn=0-19-852011-5}}</ref>
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| we can write
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| :<math>\langle x, \varphi \rangle = \int \langle x , y \rangle \langle y, \varphi \rangle dy. </math>
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| Then:
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| :<math>\begin{align}
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| \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\\
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| &= \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
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| \end{align}</math>
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| The function ''G(x, y; λ)'' defined by:
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| :<math>\begin{align}
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| G(x, y; \lambda) &= \left \langle x, \frac{y}{\lambda I - L} \right \rangle \\
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| &= \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 \\
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| &= \sum_{i=1}^n \frac{\langle x, e_i \rangle \langle f_i , y\rangle }{\lambda - \lambda_i} \\
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| &= \sum_{i=1}^n \frac{e_i (x) f_i^*(y) }{\lambda - \lambda_i},
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| \end{align}</math>
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| 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 |isbn=0-486-66444-9 |url=http://books.google.com/books?id=gnQeAQAAIAAJ&dq=intitle:applied+intitle:mathematics+inauthor:Friedman&cd=1 }}</ref>
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| :<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>
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| ==Operator equations==
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| {{See also|Spectral theory of ordinary differential equations|Integral equation}}
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| Consider the operator equation:
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| :<math>(O-\lambda I ) |\psi \rangle = |h \rangle; </math>
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| in terms of coordinates:
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| :<math>\int \langle x, (O-\lambda I)y \rangle \langle y, \psi \rangle dy = h(x). </math>
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| A particular case is λ = 0.
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| The Green's function of the previous section is:
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| :<math>\langle y, G(\lambda) z\rangle = \left \langle y, (O-\lambda I)^{-1} z \right \rangle = G(y, z; \lambda),</math>
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| and satisfies:
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| :<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>
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| Using this Green's function property:
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| :<math>\int \langle x, (O-\lambda I) y \rangle G(y, z; \lambda ) dy = \delta (x-z). </math>
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| Then, multiplying both sides of this equation by ''h''(''z'') and integrating:
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| :<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>
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| which suggests the solution is:
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| :<math>\psi(x) = \int h(z) G(x, z; \lambda) dz.</math>
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| 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:
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| :<math>G(x, z; \lambda) = \sum_{i=1}^n \frac{e_i (x) f_i^*(z)}{\lambda - \lambda_i}.</math>
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| There are many other ways to find ''G'', of course.<ref name=Green>
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| 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 |url=http://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=http://books.google.com/books?id=UUfy8CcJiDkC&printsec=frontcover|isbn=0-08-045134-9 |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#Inhomogenous_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.
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| ==Spectral theorem and Rayleigh quotient==
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| [[Optimization problem]]s may be the most useful examples about the combinatorial significance of the eigenvalues and eigenvectors in symmetric matrices, especially for the [[Rayleigh quotient]] with respect to a matrix '''M'''.
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| '''Theorem''' ''Let '''M''' be a symmetric matrix and let '''x''' be the non-zero vector that maximize the [[Rayleigh quotient]] with respect to '''M'''. Then, '''x''' is an eigenvector of '''M''' with eigenvalue equal to the [[Rayleigh quotient]]. Moreover, this eigenvalue is the largest eigenvalue of '''M'''. ''
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| '''Proof''' Assume the spectral theorem. let the eigenvalues of '''M''' are <math>\lambda_1\le\lambda_2\le\cdots\le\lambda_n</math>. Since '''{<math>v_i</math>}i''' is an [[orthonormal basis]], any vector x in this basis is
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| <math>x = \sum_{i}\ x\ v_{i}^{T} \ v_{i}</math>
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| The way to prove this formula is pretty easy.
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| :<math>v_j^{T}\sum_{i} v_i^{T} x v_i</math>
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| :<math> = \sum_{i} v_i^{T} x v_j^{T} v_i</math>
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| :<math> = (v_j^{T} x ) v_j^{T} v_j</math>
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| :<math> = v_j^{T} x</math>
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| evaluate the [[Rayleigh quotient]] with respect to x:
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| :<math>\frac{x^{T} M x}{x^{T} x}</math>
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| :<math>= (\sum_{i} (v_i^{T} x) v_i)^{T} M (\sum_{j} (v_j^{T} x) v_j)</math>
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| :<math>= (\sum_{i} (v_i^{T} x) v_i)^{T}) (\sum_{j} (v_j^{T} x) v_j\lambda_j)</math>
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| :<math>= \sum_{i,j} (v_i^{T} x) v_i)^{T})(v_j^{T} x) v_j\lambda_j)</math>
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| :<math>= \sum_{j} (v_j^{T} x)(v_j^{T} x)\lambda_j</math>
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| :<math>= \sum_{j} (v_j^{T} x)^2\lambda_j\le\lambda_n \sum_{j} (v_j^{T} x)^2</math>
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| :<math>= \lambda_n </math>
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| so the [[Rayleigh quotient]] is always less than <math>\lambda_n</math>.
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| <ref> Spielman,Daniel A. "Lecture Note of Spectral Graph Theory" Yale University(2012) http://cs.yale.edu/homes/spielman/561/ .</ref>
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| ==See also==
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| * [[Spectrum (functional analysis)]], [[Resolvent formalism]], [[Decomposition of spectrum (functional analysis)]]
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| * [[Spectral radius]], [[Spectrum of an operator]], [[Spectral theorem]]
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| * [[Self-adjoint operator]], [[functional calculus|Functions of operators]], [[Operator theory]]
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| * [[Sturm–Liouville theory]], [[Integral equation]]s, [[Fredholm theory]]
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| * [[Compact operator]]s, [[Isospectral]] operators, [[Completeness]]
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| * [[Lax pair]]s
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| * [[Spectral geometry]]
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| * [[Spectral graph theory]]
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| * [[List of functional analysis topics]]
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| ==Notes==
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| {{Reflist}}
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| ==General references==
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| * {{Cite book |title=Spectral Theory and Differential Operators; ''Volume 42 in the ''Cambridge Studies in Advanced Mathematics |author=Edward Brian Davies |publisher=Cambridge University Press |year=1996 |url=http://books.google.com/books?id=EXtKuJAksSUC&printsec=frontcover&dq=intitle:Spectral+intitle:Theory+intitle:and+intitle:Differential+intitle:Operators&cd=1#v=onepage&q=Spectral%20theory%20&f=false |isbn=0-521-58710-7 }}
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| * {{Cite book |title=Linear Operators, Spectral Theory, Self Adjoint Operators in Hilbert Space (Part 2) |author= Nelson Dunford; Jacob T Schwartz |publisher=Wiley |year=1988 |url=http://books.google.com/books?id=eOFfQQAACAAJ&dq=isbn:0471608475&cd=1 |isbn=0-471-60847-5 |edition=Paperback reprint of 1967 }}
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| *{{Cite book |title=Linear Operators, Spectral Operators (Part 3) |author= Nelson Dunford; Jacob T Schwartz |publisher=Wiley |year=1988 |isbn=0-471-60846-7 |url=http://books.google.com/books?id=B0SeJNIh3BwC&printsec=frontcover&dq=isbn:0471608467&cd=1#v=onepage&q=&f=false |edition=Paperback reprint of 1971}}
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| * {{Cite book |title=Mathematical Physics: a Modern Introduction to its Foundations |author= Sadri Hassani |publisher=Springer |year=1999 |url=http://books.google.com/books?id=BCMLOp6DyFIC&pg=RA1-PA109#v=onepage&q=&f=false|chapter=Chapter 4: Spectral decomposition |isbn=0-387-98579-4 }}
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| *{{Springer|id=S/s086520|title=Spectral theory of linear operators}}
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| * {{Cite book |title=Spectral Theory of Banach Space Operators; |author=Shmuel Kantorovitz|year=1983 | publisher= Springer}}
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| * {{Cite book |title=Linear Operator Theory in Engineering and Science; ''Volume 40 of Applied mathematical sciences'' |author=Arch W. Naylor, George R. Sell |page=411 |chapter=Chapter 5, Part B: The Spectrum |url=http://books.google.com/books?id=t3SXs4-KrE0C&pg=PA411&dq=%22resolution+of+the+identity%22&cd=8#v=onepage&q=%22resolution%20of%20the%20identity%22&f=false |isbn=0-387-95001-X |year=2000 |publisher=Springer}}
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| * {{Cite book |title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators |author= Gerald Teschl |authorlink= Gerald Teschl |publisher= American Mathematical Society |year=2009 |url=http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=978-0-8218-4660-5 }}
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| * {{Cite book |title=Spectral Theory and Quantum Mechanics; With an Introduction to the Algebraic Formulation |author= Valter Moretti |authorlink= Valter Moretti |publisher= Springer |year=2013 |url=http://www.springer.com/mathematics/applications/book/978-88-470-2834-0|isbn=978-88-470-2834-0 }}
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| ==External links==
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| *[http://www.mathphysics.com/opthy/OpHistory.html Evans M. Harrell II]: A Short History of Operator Theory
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| *{{cite journal |doi=10.1006/hmat.1995.1025 |author=Gregory H. Moore |title= The axiomatization of linear algebra: 1875-1940 |journal=Historia Mathematica |year=1995 |volume=22 |pages=262–303}}
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Spectral Theory}}
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| [[Category:Linear algebra]]
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| [[Category:Spectral theory|*]]
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| [[de:Spektrum_(Operatortheorie)]]
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