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{{Short description|Construction in functional analysis, useful to solve differential equations}}
The [[spectrum (functional analysis)|spectrum]] of a [[linear operator]] <math>T</math> that operates on a [[Banach space]] <math>X</math>
* a '''point spectrum''', consisting of the [[eigenvalues and eigenvectors|eigenvalues]] of <math>T</math>;
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=== For bounded Banach space operators ===
Let ''X'' be a [[Banach space]], ''B''(''X'') the family of [[bounded operator]]s on ''X'', and {{math|''T''
If {{math|''T''
#{{math|''T''
#{{math|''T''
#{{math|''T''
So ''σ''(''T'') is the disjoint union of these three sets,
<math display="block">\sigma(T) = \sigma_p (T) \cup \sigma_c (T) \cup \sigma_r (T).</math>The complement of the spectrum <math>\sigma(T)</math> is known as [[resolvent set]] <math>\rho(T)</math> that is <math>\rho(T)=\mathbb{C}\setminus\sigma(T)</math>.
{| class="wikitable"
!rowspan=2| Surjectivity of {{math|''T'' − ''λ''}} !!colspan=3| Injectivity of {{math|''T'' − ''λ''}}
|-
! Injective and bounded below !! Injective but not bounded below !! not injective
|-
! Surjective
| Resolvent set {{math|''ρ''(''T'')}} || Nonexistent ||rowspan=3| Point spectrum {{math|''σ<sub>p</sub>''(''T'')}}
|-
! Not surjective but has dense range
| Nonexistent || Continuous spectrum {{math|''σ<sub>c</sub>''(''T'')}}
|-
! Does not have dense range
|colspan=2| Residual spectrum {{math|''σ<sub>r</sub>''(''T'')}}
|}
In addition, when {{math|''T'' − ''λ''}} does not have dense range, whether is injective or not, then ''λ'' is said to be in the '''compression spectrum''' of ''T'', ''σ<sub>cp</sub>''(''T''). The compression spectrum consists of the whole residual spectrum and part of point spectrum.
=== 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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==== Multiplication operator ====
Given a σ-finite [[measure space]] (''S'', ''Σ'', ''μ''), consider the Banach space [[Lp space|''L<sup>p</sup>''(''
<math display="block">(T_h f)(s) = h(s) \cdot f(s).</math>
The operator norm of ''T'' is the essential supremum of ''h''. The [[essential range]] of ''h'' is defined in the following way: a complex number ''λ'' is in the essential range of ''h'' if for all ''ε'' > 0, the preimage of the open ball ''B<sub>ε</sub>''(''λ'') under ''h'' has strictly positive measure. We will show first that ''σ''(''T<sub>h</sub>'') coincides with the essential range of ''h'' and then examine its various parts.
If ''λ'' is not in the essential range of ''h'', take ''ε'' > 0 such that ''h''<sup>
''
<math display="block">
\| (T_h - \lambda) f_n \|_p ^p = \| (h - \lambda) f_n \|_p ^p = \int_{S_n} | h - \lambda \; |^p d \mu
\leq \frac{1}{n^p} \; \mu(S_n) = \frac{1}{n^p} \| f_n \|_p ^p.
</math>
This shows {{math|''T<sub>h</sub>''
If ''λ'' is such that ''μ''( ''h''<sup>−1</sup>({''λ''})) > 0, then ''λ'' lies in the point spectrum of ''T<sub>h</sub>'' as follows. Let ''f'' be the characteristic function of the measurable set ''h''<sup>−1</sup>(''λ''), then by considering two cases, we find
<math display="block">\forall s \in S, \; (T_h f)(s) = \lambda f(s),</math>
so λ is an eigenvalue of ''T''<sub>''h''</sub>.
Any ''λ'' in the essential range of ''h'' that does not have a positive measure preimage is in the continuous spectrum of ''T<sub>h</sub>''. To show this, we must show that {{math|''T<sub>h</sub>''
<math display="block">f_n(s) = \frac{1}{ h(s) - \lambda} \cdot g_n(s) \cdot f(s).</math>
Direct calculation shows that ''f<sub>n</sub>'' ∈ ''L<sup>p</sup>''(''μ''), with <math>\| f_n\|_p\leq n \|f\|_p</math>. Then by the [[dominated convergence theorem]],
<math display="block">(T_h - \lambda) f_n \rightarrow f</math>
in the ''L<sup>p</sup>''(''μ'') norm.
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{{main|Shift space}}
In the special case when ''S'' is the set of natural numbers and ''μ'' is the counting measure, the corresponding ''L<sup>p</sup>''(''μ'') is denoted by l<sup>''p''</sup>. This space consists of complex valued sequences {''x<sub>n</sub>''} such that
<math display="block">\sum_{n \geq 0} | x_n |^p < \infty.</math>
For 1 < ''p'' < ∞, ''l <sup>p</sup>'' is [[Reflexive space|reflexive]]. Define the [[shift operator|left shift]] ''T'' : ''l <sup>p</sup>'' → ''l <sup>p</sup>'' by
<math display="block">T(x_1, x_2, x_3, \dots) = (x_2, x_3, x_4, \dots).</math>
''T'' is a [[partial isometry]] with operator norm 1. So ''σ''(''T'') lies in the closed unit disk of the complex plane.
''T*'' is the right shift (or [[unilateral shift]]), which is an isometry on ''l <sup>q</sup>'', where 1/''p'' + 1/''q'' = 1:
<math display="block">T^*(x_1, x_2, x_3, \dots) = (0, x_1, x_2, \dots).</math>
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 in [[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
<math display="block">T x = \lambda x, \qquad i.e. \; (x_2, x_3, x_4, \dots) = \lambda (x_1, x_2, x_3, \dots),</math>
then
<math display="block">x = x_1 (1, \lambda, \lambda^2, \dots),</math>
which cannot be in ''l <sup>p</sup>'', a contradiction. This means the unit circle must lie in the continuous spectrum of ''T''.
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{{further information|Borel functional calculus}}
This subsection briefly sketches the development of this calculus. The idea is to first establish the continuous functional calculus, and then pass to measurable functions via the [[
# If ''T'' is self-adjoint, then for any polynomial ''P'', the operator norm satisfies <math display="block">\| P(T) \| = \sup_{\lambda \in \sigma(T)} |P(\lambda)|.</math>
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For a fixed ''h'' ∈ ''H'', we notice that
<math display="block">f \rightarrow \langle h, f(T) h \rangle</math>
is a positive linear functional on ''C''(''σ''(''T'')). According to the Riesz–Markov–Kakutani representation theorem a unique measure ''μ<sub>h</sub>'' on ''σ''(''T'') exists such that
<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''. 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>
Via the [[polarization identity]], one can recover (since ''H'' is assumed to be complex)
<math display="block">\langle k, g(T) h \rangle.</math>
and therefore ''g''(''T'') ''h'' for arbitrary ''h''.
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Let ''h'' ∈ ''H'' and ''μ<sub>h</sub>'' be its corresponding spectral measure on ''σ''(''T'') ⊂ '''R'''. According to a refinement of [[Lebesgue's decomposition theorem]], ''μ<sub>h</sub>'' can be decomposed into three mutually singular parts:
where ''μ''<sub>ac</sub> is absolutely continuous with respect to the Lebesgue measure, ''μ''<sub>sc</sub> is singular with respect to the Lebesgue measure and atomless, and ''μ''<sub>pp</sub> is a pure point measure.{{sfn|Simon|2005|page=43}}{{sfn | Teschl | 2014 | p=114-119}}
All three types of measures are invariant under linear operations. Let ''H''<sub>ac</sub> be the subspace consisting of vectors whose spectral measures are absolutely continuous with respect to the [[Lebesgue measure]]. Define ''H''<sub>pp</sub> and ''H''<sub>sc</sub> in analogous fashion. These subspaces are invariant under ''T''. For example, if ''h'' ∈ ''H''<sub>ac</sub> and ''k'' = ''T h''. Let ''χ'' be the characteristic function of some Borel set in ''σ''(''T''), then
<math display="block">\langle k, \chi(T) k \rangle = \int_{\sigma(T)} \chi(\lambda) \cdot \lambda^2 d \mu_{h}(\lambda) = \int_{\sigma(T)} \chi(\lambda) \; d \mu_k(\lambda).</math>
So
<math display="block">\lambda^2 d \mu_{h} = d \mu_{k}</math>
and ''k'' ∈ ''H''<sub>ac</sub>. Furthermore, applying the spectral theorem gives
<math display="block">H = H_{\mathrm{ac}} \oplus H_{\mathrm{sc}} \oplus H_{\mathrm{pp}}.</math>
This leads to the following definitions:
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#The set of eigenvalues of ''T'' is called the '''pure point spectrum''' of ''T'', ''σ''<sub>pp</sub>(''T'').
The closure of the eigenvalues is the spectrum of ''T'' restricted to ''H''<sub>pp</sub>.{{sfn|Simon|2005|page=44}}<ref
So
=== Comparison ===
A bounded self-adjoint operator on Hilbert space is, a fortiori, a bounded operator on a Banach space. Therefore, one can also apply to ''T'' the decomposition of the spectrum that was achieved above for bounded operators on a Banach space. Unlike the Banach space formulation,{{clarify|reason=Why should they be related? The definition is quite different.|date=May 2015}} the union
<math display="block">\sigma(T) = {\bar \sigma_{\mathrm{pp}}(T)} \cup \sigma_{\mathrm{ac}}(T) \cup \sigma_{\mathrm{sc}}(T)</math>
need not be disjoint. It is disjoint when the operator ''T'' is of uniform multiplicity, say ''m'', i.e. if ''T'' is unitarily equivalent to multiplication by ''λ'' on the direct sum
<math display="block">\bigoplus _{i = 1} ^m L^2(\mathbb{R}, \mu_i)</math>
for some Borel measures <math>\mu_i</math>. When more than one measure appears in the above expression, we see that it is possible for the union of the three types of spectra to not be disjoint. If {{math|''λ'' ∈ ''σ<sub>ac</sub>''(''T'') ∩ ''σ<sub>pp</sub>''(''T'')}}, ''λ'' is sometimes called an eigenvalue ''embedded'' in the absolutely continuous spectrum.
When ''T'' is unitarily equivalent to multiplication by ''λ'' on
<math display="block">L^2(\mathbb{R}, \mu),</math>
the decomposition of ''σ''(''T'') from Borel functional calculus is a refinement of the Banach space case.
===
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
The [[Self-adjoint_operator#Pure_point_spectrum|pure point spectrum]] corresponds to [[bound state]]s in the following way:
* 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}}
* 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}}
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
:<math> f(x) = \begin{cases}
n & \text{if }x \in \left[n, n+\frac{1}{n^4}\right] \\
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>
Known 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 quantum mechanical measurements, one encounters "[[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'''). These are [[Free_particle|free states]] belonging to the absolutely continuous spectrum. In the [[Spectral_theorem#Unbounded_self-adjoint_operators|spectral theorem for unbounded self-adjoint operators]], these states are referred to as "generalized eigenvectors" of an observable with "generalized eigenvalues" that do not necessarily belong to its 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.
The singular spectrum correspond to physically impossible outcomes. It was believed for some time that the singular spectrum
== Decomposition into essential spectrum and discrete spectrum ==
Let <math>A:\,X\to X</math> be a closed operator defined on the domain <math>D(A)\subset X</math> which is dense in ''X''. Then there is a decomposition of the spectrum of ''A'' into a [[disjoint union]],{{sfn | Teschl | 2014 | p=170}}
<math display="block">\sigma(A)=\sigma_{\mathrm{ess},5}(A)\sqcup\sigma_{\mathrm{d}}(A),</math>
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
== See also ==
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* [[Spectrum (functional analysis)]]
==Notes==
{{Reflist|group=nb}}
{{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}}
{{Functional analysis}}
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