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=== For unbounded operators ===
The spectrum of an [[unbounded operator]] can be divided into three parts in the same way as in the bounded case, but because the operator is not defined everywhere, the definitions of domain, inverse, etc. are more involved.
 
=== Examples ===
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For ''λ'' ∈ '''C''' with |''λ''| < 1,
<math display="block">x = (1, \lambda, \lambda ^2, \dots) \in l^p</math>
and ''T x'' = ''λ x''. Consequently, the point spectrum of ''T'' contains the open unit disk. Now, ''T*'' has no eigenvalues, i.e. ''σ<sub>p</sub>''(''T*'') is empty. Thus, invoking reflexivity and the theorem givenin above[[Spectrum_(functional_analysis)#Spectrum_of_the_adjoint_operator]] (that ''σ<sub>p</sub>''(''T'') ⊂ ''σ<sub>r</sub>''(''T''*) ∪ ''σ<sub>p</sub>''(''T''*)), we can deduce that the open unit disk lies in the residual spectrum of ''T*''.
 
The spectrum of a bounded operator is closed, which implies the unit circle, { |''λ''| = 1 } ⊂ '''C''', is in ''σ''(''T''). Again by reflexivity of ''l <sup>p</sup>'' and the theorem given above (this time, that {{math|''σ<sub>r</sub>''(''T'') ⊂ ''σ<sub>p</sub>''(''T''*)}}), we have that ''σ<sub>r</sub>''(''T'') is also empty. Therefore, for a complex number ''λ'' with unit norm, one must have ''λ'' ∈ ''σ<sub>p</sub>''(''T'') or ''λ'' ∈ ''σ<sub>c</sub>''(''T''). Now if |''λ''| = 1 and
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<math display="block">\int_{\sigma(T)} f \, d \mu_h = \langle h, f(T) h \rangle.</math>
 
This measure is sometimes called the '''[[spectral measure]]''' associated to h'' h''. The spectral measures can be used to extend the continuous functional calculus to bounded Borel functions. For a bounded function ''g'' that is Borel measurable, define, for a proposed ''g''(''T'')
<math display="block">\int_{\sigma(T)} g \, d \mu_h = \langle h, g(T) h \rangle.</math>
 
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the decomposition of ''σ''(''T'') from Borel functional calculus is a refinement of the Banach space case.
 
=== Quantum physicsmechanics ===
 
The preceding comments can be extended to the unbounded self-adjoint operators since Riesz-Markov holds for [[locally compact]] [[Hausdorff space]]s.
 
In [[mathematical formulation of quantum mechanics|quantum mechanics]], observables are (often unbounded) [[self-adjoint operator]]s, often not bounded, and their spectra are the possible outcomes of measurements. Absolutely continuous spectrum of a physical observable corresponds to [[Free_particle|free states]] of a system, while the [[Self-adjoint_operator#Pure_point_spectrum|pure point spectrum]] corresponds to [[bound state]]s. The singular spectrum correspond to physically impossible outcomes. An example of a quantum mechanical observable which has purely continuous spectrum is the [[position operator]] of a free particle moving on a line. Its spectrum is the entire real line. Also, since the [[momentum operator]] is unitarily equivalent to the position operator, via the [[Fourier transform]], they have the same spectrum.
 
The [[Self-adjoint_operator#Pure_point_spectrum|pure point spectrum]] corresponds to [[bound state]]s in the following way:
Intuition may induce one to say that the discreteness of the spectrum is intimately related to the corresponding states being "localized". However, a careful mathematical analysis shows that this is not true. The following <math>f</math> is an element of <math>L^2(\mathbb{R})</math> and increasing as <math>x \to \infty</math>.
* A [[quantum state]] is a bound state if and only if it is finitely [[Probability amplitude#Normalization|normalizable]] for all times <math>t\in\mathbb{R}</math>.{{sfn | Ruelle | 1969}}
<math display="block"> f(x) = \begin{cases}
* An observable has pure point spectrum if and only if its [[Quantum_state#Eigenstates_and_pure_states|eigenstates]] form an [[orthonormal basis]] of <math>H</math>.{{sfn | Simon | 1978|p=3}}
n & \text{if }x \in \left[n, n+\frac{1}{n^4}\right], \\▼
A particle is said to be in a bound state if it remains "localized" in a bounded region of space.{{sfn | Blanchard | Brüning | 2015 | p=430}} Intuitively one might therefore think that the "discreteness" of the spectrum is intimately related to the corresponding states being "localized". However, a careful mathematical analysis shows that this is not true in general.{{sfn | Blanchard | Brüning | 2015 | p=432}} For example, consider the function
0 & \text{else.}▼
:<math> f(x) = \endbegin{cases} </math>
▲n & \text{if }x \in \left[n, n+\frac{1}{n^4}\right], \\
However, the phenomena of [[Anderson localization]] and [[dynamical localization]] describe, when the eigenfunctions are localized in a physical sense. Anderson Localization means that eigenfunctions decay exponentially as <math> x \to \infty </math>. Dynamical localization is more subtle to define.▼
▲0 & \text{else.}
\end{cases}, \quad \forall n \in \mathbb{N}. </math>
This function is normalizable (i.e. <math>f\in L^2(\mathbb{R})</math>) as
:<math>\int_{n}^{n+\frac{1}{n^4}}n^2\,dx = \frac{1}{n^2} \Rightarrow \int_{-\infty}^{\infty} |f(x)|^2\,dx = \sum_{n=1}^\infty \frac{1}{n^2}.</math>
▲HoweverKnown as the [[Basel problem]], this series converges to <math display="inline">\frac{\pi^2}{6}</math>. Yet, <math>f</math> increases as <math>x \to \infty</math>, i.e, the state "escapes to infinity". The phenomena of [[Anderson localization]] and [[dynamical localization]] describe, when the eigenfunctions are localized in a physical sense. Anderson Localization means that eigenfunctions decay exponentially as <math> x \to \infty </math>. Dynamical localization is more subtle to define.
 
Sometimes, when performing physical quantum mechanical calculationsmeasurements, one encounters "eigenvectors[[Quantum_state#Eigenstates_and_pure_states|eigenstates]]" that are not localized, e.g., quantum states that do not lie in ''L''<sup>2</sup>('''R'''), i.e. waveThese functionsare that[[Free_particle|free arestates]] notbelonging localized.to the Theseabsolutely arecontinuous spectrum. In the free[[Spectral_theorem#Unbounded_self-adjoint_operators|spectral theorem for unbounded self-adjoint operators]], these states ofare thereferred system.to as As"generalized statedeigenvectors" above,of inan theobservable mathematicalwith formulation,"generalized theeigenvalues" freethat statesdo correspondnot tonecessarily thebelong absolutelyto continuousits spectrum. Alternatively, if it is insisted that the notion of eigenvectors and eigenvalues survive the passage to the rigorous, one can consider operators on [[rigged Hilbert space]]s.{{sfn|de la Madrid Modino|2001|pp=95-97}}
 
An example of an observable whose spectrum is purely absolutely continuous is the [[position operator]] of a free particle moving on the entire real line. Also, since the [[momentum operator]] is unitarily equivalent to the position operator, via the [[Fourier transform]], it has a purely absolutely continuous spectrum as well.
It was believed for some time that singular spectrum is something artificial. However, examples as the [[almost Mathieu operator]] and [[random Schrödinger operator]]s have shown, that all types of spectra arise naturally in physics.▼
 
▲The singular spectrum correspond to physically impossible outcomes. It was believed for some time that the singular spectrum iswas something artificial. However, examples as the [[almost Mathieu operator]] and [[random Schrödinger operator]]s have shown, that all types of spectra arise naturally in physics.{{sfn | Jitomirskaya | Simon | 1994}}{{sfn | Simon | Stolz | 1996}}
 
== Decomposition into essential spectrum and discrete spectrum ==
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where
# <math>\sigma_{\mathrm{ess},5}(A)</math> is the fifth type of the [[essential spectrum]] of ''A'' (if ''A'' is a [[self-adjoint operator]], then <math>\sigma_{\mathrm{ess},k}(A)=\sigma_{\mathrm{ess}}(A)</math> for all <math>1\le k\le 5</math>);
# <math>\sigma_{\mathrm{d}}(A)</math> is the [[discrete spectrum (mathematics)|discrete spectrum]] of ''A'', which consists of [[normal eigenvalue]]s, or, equivalently, of [[isolated point]]s of <math>\sigma(A)</math> such that the corresponding [[Riesz projector]] has a finite rank. It is a a proper subset of the [[point spectrum]], i.e., <math>\sigma_d(A)\subset\sigma_p(A)</math>, as the set of eigenvalues of ''A'' need not necessarily be isolated points of the spectrum.
 
== See also ==
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{{Reflist}}
== References ==
* {{cite book | last=Blanchard | first=Philippe | last2=Brüning | first2=Erwin | title=Mathematical Methods in Physics | publisher=Birkhäuser | date=2015 | isbn=978-3-319-14044-5}}
* {{cite book | last=Dunford | first=N. | last2=Schwartz | first2=J. T. | title=Linear Operators, Part 1: General Theory | publisher=John Wiley & Sons | year=1988 | isbn=0-471-60848-3}}
* {{cite journal | last=Jitomirskaya | first=S. | last2=Simon | first2=B. | title=Operators with singular continuous spectrum: III. Almost periodic Schrödinger operators | journal=Communications in Mathematical Physics | volume=165 | issue=1 | date=1994 | issn=0010-3616 | doi=10.1007/BF02099743 | pages=201–205}}
*{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree= PhD |publisher= Universidad de Valladolid}}
* {{cite book | last=Reed | first=M. | last2=Simon | first2=B. | title=Methods of Modern Mathematical Physics: I: Functional analysis | publisher=Academic Press | year=1980 | isbn=978-0-12-585050-6 }}
* {{cite journal | last=Ruelle | first=D. | title=A remark on bound states in potential-scattering theory | journal=Il Nuovo Cimento A | publisher=Springer Science and Business Media LLC | volume=61 | issue=4 | year=1969 | issn=0369-3546 | doi=10.1007/bf02819607 | url=https://www.ihes.fr/%7Eruelle/PUBLICATIONS/%5B25%5D.pdf}}
* {{cite web | last=Simon | first=B. | title=An Overview of Rigorous Scattering Theory | date=1978 | url=https://api.semanticscholar.org/CorpusID:16913591}}
* {{cite journal | last=Simon | first=B. | last2=Stolz | first2=G. | title=Operators with singular continuous spectrum, V. Sparse potentials | journal=Proceedings of the American Mathematical Society | volume=124 | issue=7 | date=1996 | issn=0002-9939 | doi=10.1090/S0002-9939-96-03465-X | pages=2073–2080| doi-access=free }}
* {{cite book | last1=Simon | first1=Barry | author1-link=Barry Simon | title=Orthogonal polynomials on the unit circle. Part 1. Classical theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-3446-6 | mr=2105088 |year=2005 | volume=54}}
* {{cite book | last=Teschl | first=G. | title=Mathematical Methods in Quantum Mechanics | publisher=American Mathematical Soc. | publication-place=Providence (R.I) | date=2014 | isbn=978-1-4704-1704-8}}
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