查看“︁谱理论”︁的源代码

[[数学]]中，'''谱理论'''是将单一方阵的[[特征向量和特征值]]理论推广到各种[[空间 (数学)|空间]]中[[算子]]结构的更广泛理论的统称，<ref name="Dieudonné">{{cite book |title=History of functional analysis |author=Jean Alexandre Dieudonné |url=https://books.google.com/books?id=mg7r4acKgq0C |isbn=0-444-86148-3 |year=1981 |publisher=Elsevier}}</ref>是[[线性代数]]和[[线性方程组]]解及其推广的成果。<ref name=Arveson>{{cite book |title=A short course on spectral theory |author=William Arveson |chapter=Chapter 1: spectral theory and Banach algebras |url =https://books.google.com/books?id=ARdehHGWV1QC |isbn=0-387-95300-0 |year=2002 |publisher=Springer}}</ref>谱理论与[[解析函数]]理论相关，因为算子的谱特性与谱参数的解析函数相关。<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 |chapter-url=https://books.google.com/books?id=SR1QkG6OkVEC&pg=PA181 |page= 181 ''et seq'' |isbn=0-306-11028-8 |year=1991 |publisher=Springer}}
</ref>

==数学背景==
[[大卫·希尔伯特]]在[[希尔伯特空间]]理论的最初表述中首先引入了“谱理论”，以无穷多变量的[[二次型]]为基础。因此，最初的[[谱定理]]被认为是[[椭球]][[主轴定理]]在无穷维中的形式。后来，[[量子力学]]中发现谱理论可解释原子[[发射光谱]]特征纯粹是偶然的。希尔伯特自己也对这出乎意料的应用感到惊讶：“我从纯粹的数学兴趣出发，发展了无穷多变量理论，甚至把它称作‘谱分析’，却从没想到它能用在物理学中真正的光谱上。”<ref>{{cite web|last1=Steen|first1=Lynn Arthur|title=Highlights in the History of Spectral Theory|url=http://www.stolaf.edu/people/steen/Papers/73spectral.pdf|website=St. Olaf College|access-date=2015-12-14|url-status=dead|archive-url=https://web.archive.org/web/20160304050120/http://www.stolaf.edu/people/steen/Papers/73spectral.pdf|archive-date=2016-03-04}}</ref>

谱理论有3种表述，分别用于不同领域。在希尔伯特的最初表述后，抽象[[希尔伯特空间]]及其上单[[正规算子]]的谱理论很好地满足了物理学的要求，冯诺依曼的工作就是例子。<ref name= vonNeumann>

{{Cite book |title=The mathematical foundations of quantum mechanics; ''Volume 2 in Princeton'' Landmarks in Mathematics ''series'' |author=John von Neumann |url=https://books.google.com/books?id=JLyCo3RO4qUC&q=mathematical+foundations+of+quantum+mechanics+inauthor:von+inauthor:neumann |isbn=0-691-02893-1 |year=1996 |publisher =Princeton University Press |edition=Reprint of translation of original 1932 }}

</ref>在此基础上，进一步发展的理论涉及一般[[巴拿赫代数]]。这引出了涵盖[[交换巴拿赫代数|交换]]情形的[[盖尔范德表示]]，进一步启发了[[非交换调和分析]]。

从与[[傅立叶分析]]的联系中，可看出两者之别。[[实线]]上的[[傅立叶变换]]某种意义上是[[导数]]作为[[微分算子]]的谱理论。但为涵盖这些现象，必须处理广义特征函数（如由[[装备希尔伯特空间]]）。另一方面，构建[[群代数]]也很简单，其谱可捕捉到傅立叶变换的基本特征，这可由[[庞特里亚金对偶性]]实现。

还可以研究[[巴拿赫空间]]上算子的谱性质。例如，巴拿赫空间上的[[紧算子]]有很多类似[[矩阵]]的谱性质。

==物理背景==
[[振动]]的物理学背景是这样解释的：<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 |access-date=2023-12-24 |archive-date=2010-07-16 |archive-url=https://web.archive.org/web/20100716224400/http://www.kcl.ac.uk/schools/pse/maths/research/analysis/research.html |dead-url=no }}</ref>

{{Quote|谱理论研究各种物体的局部振动，从[[化学]]中的[[原子]]与[[分子]]直到[[声波导]]中的障碍物。振动有[[頻率 (物理學)|频率]]，问题在于确定这种局部振动何时发生，以及如何计算频率。这是个非常复杂的问题，因为每个物体不仅有[[基频]]，还有复杂的[[泛音]]，在不同物体上有迥异的表现。}}

这些物理思想在技术层面上与数学毫无关系，但也有间接涉及的例子（如“[[听出鼓的形状]]”）。希尔伯特用“谱”，是源于1897年Wilhelm Wirtinger一篇关于[[希尔微分方程]]的论文，20世纪前十年，他的学生（[[埃哈德·施密特]]、[[赫尔曼·外尔]]）也采用了这一术语。埃哈德·施密特和[[里斯·弗里杰什]]从希尔伯特的思想中发展了[[希尔伯特空间]]的概念。<ref name=Young>{{Cite book |title=An introduction to Hilbert space |author=Nicholas Young |url=https://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=https://books.google.com/books?id=gqZUGMKtNuoC&q=%22thinking+geometrically+in+Hilbert%27s+%22&pg=PA50 |isbn=0-7923-6539-9 |publisher=Springer |year=2000 }}

</ref>近20年后，[[量子力学]]由[[薛定谔方程]]表述出来时，人们才将其与原子光谱联系起来。正如[[亨利·庞加莱]]指出的，人们曾怀疑其与振动的联系，但由于缺乏对[[巴耳末系]]的解释，这种联系由于简单的定量原因被否定了。<ref>Cf. [http://www.dm.unito.it/personalpages/capietto/Spectra.pdf Spectra in mathematics and in physics] {{webarchive|url=https://web.archive.org/web/20110727024805/http://www.dm.unito.it/personalpages/capietto/Spectra.pdf |date=2011-07-27 }} by [[Jean Mawhin]], p.4 and pp. 10-11.</ref>因此，量子力学发现谱理论能解释原子光谱的特征是纯粹偶然的。

==谱的定义==
{{Main|谱 (泛函分析)}}
考虑在一般[[巴拿赫空间]]上处处有定义的[[有界算子|有界线性变换]]''T''。构造变换：
<math display="block"> R_{\zeta} = \left( \zeta I -  T \right)^{-1}.</math>

其中''I''是[[恒等算子]]，ζ是[[复数 (数学)|复数]]。 ''T''的逆<math>T^{-1}</math>定义如下：
<math display="block">T T^{-1} = T^{-1} T = I. </math>

若存在逆，则称''T''正则（regular）；若不存在，称''T''奇异（singular）。

根据这些定义，''T''的[[预解集]]是所有使<math>R_\zeta</math>存在且有界的复数ζ的集合，通常表示为<math>\rho(T)</math>。''T''的谱是所有使<math>R_\zeta</math>不存在或无界的复数ζ的集合，通常表示为<math>\sigma(T)</math>。<math>\forall \zeta\in\rho(T)</math>（即<math>R_\zeta</math>作为有界算子存在处），函数<math>R_\zeta</math>称作''T''的预解。因此，''T''的谱是复平面中''T''的预解集之补。<ref name=Lorch>{{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=https://books.google.com/books?id=X3U2AAAACAAJ}}</ref>''T''的[[特征值]]都属于<math>\sigma(T)</math>，但<math>\sigma(T)</math>可能包含非特征值。<ref name= Young2>{{cite book |title=''op. cit'' |author= Nicholas Young |date= 1988-07-21 |url=https://books.google.com/books?id=_igwFHKwcyYC&pg=PA81 |page=81 |publisher= Cambridge University Press |isbn=0-521-33717-8}}</ref>

这定义适于巴拿赫空间，但当然也有其他空间，如[[拓扑向量空间]]包含了巴拿赫空间。<ref name=Wolff>{{Cite book |title=Topological vector spaces |author1=Helmut H. Schaefer| author2= Manfred P. H. Wolff |url=https://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>{{Cite book |title=Principal structures and methods of representation theory |author=Dmitriĭ Petrovich Zhelobenko |url=https://books.google.com/books?id=3TkmvZktjp8C&pg=PA24 |isbn=	0821837311 |publisher=American Mathematical Society |year=2006}}</ref>另一方面，巴拿赫空间包括[[希尔伯特空间]]，得到了最广泛的应用和最丰富的理论成果。<ref name=Lorch2>{{Cite book |title=Spectral Theory |author=Edgar Raymond Lorch |year=2003 |isbn=0-7581-7156-0 |url=https://books.google.com/books?id=X3U2AAAACAAJ |page=57 |chapter=Chapter III: Hilbert Space}}</ref>在适当的条件下，希尔伯特空间中变换的谱的结构可以有很多论述。特别是，[[自伴算子]]谱位于[[实线]]上；（一般来说）是离散特征值的点谱与连续谱的[[谱分解 (泛函分析)|谱组合]]。<ref name=Lorch3>{{Cite book |title=Spectral Theory |author=Edgar Raymond Lorch |year=2003 |isbn=0-7581-7156-0 |url=https://books.google.com/books?id=X3U2AAAACAAJ |page=106 ''ff'' |chapter=Chapter V: The Structure of Self-Adjoint Transformations}}</ref>

==谱理论简介==
{{Main|谱定理}}
{{See also|特征值和特征向量}}
[[泛函分析]]和[[线性代数]]中，谱定理规定了一些条件，之下算子可用简单的形式表为更简单算子之和。由于本文不适合进行全面严谨的叙述，我们采用另一种方法，避免了难以理解的形式化。

要描述这主题，最简单的方法是引入表示算子的[[狄拉克符号]]。<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=https://books.google.com/books?id=gnQeAQAAIAAJ&q=intitle:applied+intitle:mathematics+inauthor:Friedman |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=https://books.google.com/books?id=XehUpGiM6FIC&pg=PA29}}</ref>例如，非常特殊的线性算子''L''可写为[[并矢张量]]积：<ref name=Audretsch>{{Cite book |title=Entangled systems: new directions in quantum physics |author=Jürgen Audretsch |page=5 |chapter-url=https://books.google.com/books?id=8NxIgwAOU6IC&pg=PA5 |chapter=Chapter 1.1.2: Linear operators on the Hilbert space |isbn=978-3-527-40684-5 |publisher=Wiley-VCH |year=2007}}</ref><ref name=Howland>{{Cite book |title=Intermediate dynamics: a linear algebraic approach |url=https://books.google.com/books?id=SepP8-W3M0AC&q=dyad+representation+operator&pg=PA69 |page=69 ''ff'' |author=R. A. Howland |publisher=Birkhäuser |year=2006 |isbn=0-387-28059-6 |edition=2nd}}</ref>

:<math> L = | k_1 \rangle \langle b_1 |, </math>

用“左括号”<math>\langle b_1|</math>和“右括号”<math>|k_1\rangle</math>表示。定义在坐标<math>(x_1, x_2, x_3, \dots)</math>上的函数<math>f(x)</math>表示为
:<math> f(x)=\langle x, f\rangle </math>
''f''的振幅用下式表示
:<math> \|f \|^2 = \langle f, f\rangle =\int \langle f, x\rangle \langle x, f \rangle \, dx = \int f^*(x) f(x) \, dx </math>
符号(*)表示[[共轭复数]]。这种[[内积]]选择定义了一个非常特殊的[[内积空间]]，限制了后面论证的一般性。<ref name=Lorch2/>

''L''对函数''f''的影响可表述为

:<math> L | f\rangle = | k_1 \rangle \langle b_1 | f \rangle </math>

表示''L''对''f''的影响是产生新函数<math> | k_1 \rangle </math>乘以<math>\langle b_1 | f \rangle </math>所代表的内积。
 
更一般的线性算子''L''可表示为

:<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>

其中<math> \{ \, \lambda_i \, \}</math>是标量，<math> \{ \, | e_i \rangle \, \} </math>是[[基 (线性代数)|基]]，<math> \{ \, \langle f_i | \, \} </math>是[[对偶基]]。基与对偶基的关系由下式描述：

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

若这种形式化适用， 则<math> \{ \, \lambda_i \, \}</math>是''L''的[[特征值]]，函数<math> \{ \, | e_i \rangle \, \} </math>是''L''的[[特征函数]]。特征值位于''L''的谱中。<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=https://books.google.com/books?id=gnQeAQAAIAAJ&q=intitle:applied+intitle:mathematics+inauthor:Friedman}}</ref>

一些自然问题是：这种形式化在何时有效？对什么算子''L''来说，其他算子的序列展开是可能的？任何函数''f''都可用特征函数来表示（是Schauder基）吗，何时会出现点谱或连续谱？无穷维空间和有限维空间的形式化有何不同？这些观点能否推广到更一般的空间？这些问题属于谱理论范畴，需要相当的[[泛函分析]]与[[矩阵]]代数背景。

==恒同消解==
本节沿用上节的粗略表示，使用狄拉克符号，并略去严格处理的许多重要细节。<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>严谨数学处理可见参考文献。<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>特别是，空间维度''n''将是有限的。

用狄拉克符号，恒等算子可写作：

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

其中假定<math>\{ |e_i\rangle\}</math>是[[基 (线性代数)|基]]，<math> \{ \langle f_i | \}</math>是满足下式的空间的对偶基：

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

这种恒等运算表达式称作恒同的表示或消解（resolution）。<ref name= "Vujičić"/><ref name= rigor/>这种形式化表示法满足恒等的基本性质：
:<math> I^k = I </math>
对所有正整数''k''都成立。

对空间<math>| \psi \rangle</math>中的任意函数应用恒同消解，可得：

:<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>
是ψ以{&nbsp;e<sub>i</sub>&nbsp;}为基函数的广义傅立叶展开，<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 |access-date=2023-12-24 |archive-date=2023-07-26 |archive-url=https://web.archive.org/web/20230726140112/https://books.google.com/books?id=idAomhpwI8MC&pg=PA77 |dead-url=no }}</ref>
其中<math>c_i = \langle f_i | \psi \rangle.</math>

给定形式如下的算子方程：
:<math>O | \psi \rangle = | h \rangle  </math>
其中''h''在空间中，方程可在上述基之下由形式化求解：
:<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>
这将算子方程转化为[[矩阵方程]]，根据''h''的广义傅立叶系数<math>\langle f_j | h \rangle</math>和算子''O''的矩阵元素<math>O_{ji}= \langle f_j| O | e_i \rangle </math>可确定未知系数<math>c_j</math>。

谱理论可确定基与对偶基的存在性及性质。特别是，基可能由某线性算子''L''的特征函数组成：

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

其中<math>\{\lambda_i\}</math>是''L''的特征值，来自''L''的谱。那么，上述恒同消解就提供了''L''的并矢展开：

:<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>

==预解算子==
{{See also|格林函数|狄拉克δ函数}}
利用谱李理论，有预解算子''R''：

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

可据''L''的特征函数与特征值求值，并找到与''L''对应的格林函数。

将''R''应用于空间中的任意函数<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>

这函数在''L''的每个特征值处的复''λ''平面上都有[[极点 (复分析)|极点]]。于是可用[[围道积分|残差微积分]]：

:<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>

其中[[曲线积分]]是在包括''L''所有特征值的轮廓''C''上进行的。

假设函数定义在某坐标<math>\{x_j\}</math>上，即

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

引入符号

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

其中<math>\delta(x-y)=\delta(x_1-y_1,\ x_2-y_2,\ x_3-y_3,\ \ldots)</math>是[[狄拉克δ函数]]，<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>则可有
:<math>\langle x, \varphi \rangle = \int \langle x , y \rangle \langle y, \varphi \rangle dy. </math>

那么

:<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>

函数<math>G(x,\ y;\ \lambda)</math>定义如下：

:<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>

称作算子''L''的[[格林函数]]，满足：<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>

==算子方程==
{{See also|常微分方程谱理论|积分方程}}
考虑算子方程：

:<math>(O-\lambda I ) |\psi \rangle = |h \rangle; </math>

用坐标表示：

:<math>\int \langle x, (O-\lambda I)y \rangle \langle y, \psi \rangle \, dy = h(x). </math>

一个特殊情况是<math>\lambda=0</math>。

上一节的格林函数为：

:<math>\langle y, G(\lambda) z\rangle = \left \langle y, (O-\lambda I)^{-1} z \right \rangle = G(y, z; \lambda),</math>

且满足：

:<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>

利用格林函数的这一特征：

:<math>\int \langle x, (O-\lambda I) y \rangle G(y, z; \lambda ) \, dy = \delta (x-z). </math>

然后方程两边乘以<math>h(z)</math>并积分：

:<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>

这表明解是

:<math>\psi(x) = \int h(z) G(x, z; \lambda) \, dz.</math>

即，若能找到''O''的谱、构造''G''，就能找到满足算子方程的函数<math>\Psi(x)</math>，例如用

:<math>G(x, z; \lambda)  = \sum_{i=1}^n \frac{e_i (x) f_i^*(z)}{\lambda - \lambda_i}.</math>

当然有很多方法找到''G''。<ref name=Green>

例如，见{{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 |chapter-url=https://books.google.com/books?id=BCMLOp6DyFIC&pg=RA1-PA553 |year=1999 |isbn=0-387-98579-4}}、{{cite book |title=Green's function and boundary elements of multifield materials |author=Qing-Hua Qin |url=https://books.google.com/books?id=UUfy8CcJiDkC|isbn=978-0-08-045134-3 |year=2007 |publisher=Elsevier}}</ref>必须牢记的是，上述数学纯粹是形式上的，严格处理涉及相当复杂的数学，包括[[泛函分析]]、[[希尔伯特空间]]、[[分布 (数学分析)|分布]]等方面的背景知识。

==谱定理与瑞利商==
[[优化问题]]是关于对称矩阵中特征值与特征向量组合意义的重要例子，特别是关于矩阵'''M'''的[[瑞利商]]。

'''定理''' 令'''M'''为对称矩阵，'''x'''为非零向量，使对'''M'''的[[瑞利商]]最大化。则，'''x'''是'''M'''的一个特征向量，其特征值等于瑞利商；而且，该特征值是'''M'''的最大特征值。

'''证明'''  假设符合谱定理。记'''M'''的特征值为<math>\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n</math>。由于<math>\{v_i\}</math>形成[[正交基]]，所以任何向量''x''都可在此[[基 (线性代数)|基]]上表示为

: <math>x = \sum_i v_i^T x v_i</math>

证明这公式的方法非常简单。即

: <math>\begin{align}
v_j^T \sum_i v_i^T x v_i
= {} & \sum_{i} v_i^{T} x v_j^{T} v_i \\[4pt]
= {} & (v_j^T x ) v_j^T v_j \\[4pt]
= {} & v_j^T x
\end{align}</math>
求对''x''的[[瑞利商]]：

: <math>\begin{align}
x^T M x
= {} & \left(\sum_i (v_i^T x) v_i\right)^T M \left(\sum_j (v_j^T x) v_j\right) \\[4pt]
= {} & \left(\sum_i (v_i^T x) v_i^T\right) \left(\sum_j (v_j^T x) v_j\lambda_j \right) \\[4pt]
= {} & \sum_{i,j} (v_i^T x) v_i^T(v_j^T x) v_j\lambda_j \\[4pt]
= {} & \sum_j (v_j^T x)(v_j^T x)\lambda_j \\[4pt]
= {} & \sum_{j} (v_j^T x)^2\lambda_j\le\lambda_n \sum_j (v_j^T x)^2 \\[4pt]
= {} & \lambda_n x^T x,
\end{align}</math>
其中最后一行用[[帕塞瓦尔恒等式]]。最后，可得
:<math>\frac{x^T M x}{x^T x}\le \lambda_n</math>

所以瑞利商总小于<math>\lambda_n</math>。<ref>Spielman, Daniel A. "Lecture Notes on Spectral Graph Theory" Yale University (2012) http://cs.yale.edu/homes/spielman/561/ {{Wayback|url=http://cs.yale.edu/homes/spielman/561/ |date=20130504045937 }} .</ref>

==另见==
{{Portal|数学}}
* [[泛函微积分|算子函数]]、[[算子理论]]
* [[Lax_对]]
* [[最小平方频谱分析法]]
* [[自伴算子]]
* [[谱 (泛函分析)]]、[[预解形式化]]、[[谱分解 (泛函分析)]]
* [[谱 (泛函分析)]]、[[谱半径]]、[[谱定理]]
* [[施图姆-刘维尔理论]]、[[积分方程]]、[[弗雷德霍姆理论]]
* [[紧算子]]、[[等谱]]算子、[[完备空间]]
* [[谱几何]]
* [[谱图论]]

==脚注==
{{Reflist|2}}

==参考文献==
* {{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=https://books.google.com/books?id=EXtKuJAksSUC&q=Spectral+theory+ |isbn=0-521-58710-7 |access-date=2023-12-24 |archive-date=2023-04-14 |archive-url=https://web.archive.org/web/20230414195200/https://books.google.com/books?id=EXtKuJAksSUC&q=Spectral+theory+ |dead-url=no }}
* {{Cite book |title=Linear Operators, Spectral Theory, Self Adjoint Operators in Hilbert Space (Part 2) |first1= Nelson |last1=Dunford |first2=Jacob T |last2=Schwartz |publisher=Wiley |year=1988 |url=https://books.google.com/books?id=eOFfQQAACAAJ |isbn=0-471-60847-5 |edition=Paperback reprint of 1967  }}
*{{Cite book |title=Linear Operators, Spectral Operators (Part 3) |first1=Nelson |last1=Dunford |first2=Jacob T |last2=Schwartz |publisher=Wiley |year=1988 |isbn=0-471-60846-7 |url=https://books.google.com/books?id=B0SeJNIh3BwC |edition=Paperback reprint of 1971 |access-date=2023-12-24 |archive-date=2023-04-14 |archive-url=https://web.archive.org/web/20230414195201/https://books.google.com/books?id=B0SeJNIh3BwC |dead-url=no }}
* {{Cite book |title=Mathematical Physics: a Modern Introduction to its Foundations |author= Sadri Hassani |publisher=Springer |year=1999 |chapter-url=https://books.google.com/books?id=BCMLOp6DyFIC&pg=RA1-PA109|chapter=Chapter 4: Spectral decomposition |isbn=0-387-98579-4   }}
*{{Springer|id=S/s086520|title=Spectral theory of linear operators}}
* {{Cite book |title=Spectral Theory of Banach Space Operators; |author=Shmuel Kantorovitz|year=1983 | publisher= Springer}}
* {{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 |chapter-url=https://books.google.com/books?id=t3SXs4-KrE0C&q=%22resolution+of+the+identity%22&pg=PA411 |isbn=0-387-95001-X |year=2000 |publisher=Springer |access-date=2023-12-24 |archive-date=2023-12-24 |archive-url=https://web.archive.org/web/20231224083757/https://books.google.com/books?id=t3SXs4-KrE0C&q=%22resolution+of+the+identity%22&pg=PA411 |dead-url=no }}
* {{Cite book |title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators |author=Gerald Teschl |author-link=Gerald Teschl |publisher=American Mathematical Society |year=2009 |url=https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=978-0-8218-4660-5 |access-date=2023-12-24 |archive-date=2022-08-12 |archive-url=https://web.archive.org/web/20220812080117/https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |dead-url=no }}
* {{Cite book |title=Spectral Theory and Quantum Mechanics; Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation 2nd Edition |author=Valter Moretti |author-link=Valter Moretti |publisher=Springer |year=2017 |url=https://www.springer.com/it/book/9783319707051 |isbn=978-3-319-70705-1 |access-date=2023-12-24 |archive-date=2021-05-06 |archive-url=https://web.archive.org/web/20210506181954/https://www.springer.com/it/book/9783319707051 |dead-url=no }}

==外部链接==
* [http://www.mathphysics.com/opthy/OpHistory.html Evans M. Harrell II] {{Wayback|url=http://www.mathphysics.com/opthy/OpHistory.html |date=20221127150120 }}: A Short History of Operator Theory
* {{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 |issue=3 |pages=262–303|doi-access=free }}
*{{cite journal |last1=Steen |first1=L. A. |title=Highlights in the History of Spectral Theory |url=https://archive.org/details/sim_american-mathematical-monthly_1973-04_80_4/page/359 |journal=The American Mathematical Monthly |date=April 1973 |volume=80 |issue=4 |pages=359–381 |doi=10.2307/2319079|jstor=2319079 }}

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[[Category:谱理论| ]]
[[Category:线性代数]]