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''on operator perturbation''
 
''on operator perturbation''
  
The spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201301.png" /> of a (possibly unbounded) [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201302.png" /> is naturally divided into a point spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201303.png" /> (i.e., the set of eigenvalues) and a continuous spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201304.png" />, both subsets of the real axis (cf. also [[Spectrum of an operator|Spectrum of an operator]]). The continuous spectrum is very sensitive against perturbations. The Weyl–von Neumann theorem [[#References|[a2]]], Sect. X.2.2, states that for any self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201305.png" /> in a separable [[Hilbert space|Hilbert space]] there exists a (compact) self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201306.png" /> in the Hilbert–Schmidt class (cf. also [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201307.png" /> has a pure point spectrum. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201308.png" /> can be chosen arbitrarily small in the Hilbert–Schmidt norm. The set of eigenvalues obtained by perturbation of a continuous spectrum is dense in any interval covered by the continuous spectrum of the unperturbed operator. In [[Spectral theory|spectral theory]] one introduces therefore another subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w1201309.png" /> which is more stable under perturbations. The essential spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013010.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013012.png" /> is the set of all isolated eigenvalues with finite multiplicity. H. Weyl proved (in a special case) the following result [[#References|[a4]]], which is now commonly known as Weyl's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013013.png" /> be self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013014.png" /> symmetric and compact. Then
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The spectrum $\sigma ( T )$ of a (possibly unbounded) [[Self-adjoint operator|self-adjoint operator]] $T$ is naturally divided into a point spectrum $\sigma _ { p } ( T )$ (i.e., the set of eigenvalues) and a continuous spectrum $\sigma _ { c } ( T )$, both subsets of the real axis (cf. also [[Spectrum of an operator|Spectrum of an operator]]). The continuous spectrum is very sensitive against perturbations. The Weyl–von Neumann theorem [[#References|[a2]]], Sect. X.2.2, states that for any self-adjoint operator $T$ in a separable [[Hilbert space|Hilbert space]] there exists a (compact) self-adjoint operator $S$ in the Hilbert–Schmidt class (cf. also [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]]) such that $T + S$ has a pure point spectrum. Moreover, $S$ can be chosen arbitrarily small in the Hilbert–Schmidt norm. The set of eigenvalues obtained by perturbation of a continuous spectrum is dense in any interval covered by the continuous spectrum of the unperturbed operator. In [[Spectral theory|spectral theory]] one introduces therefore another subset of $\sigma ( T )$ which is more stable under perturbations. The essential spectrum $\sigma _ { \text{ess} } ( T )$ is defined as $\sigma ( T ) \backslash \sigma _ { d } ( T )$, where $\sigma _ { d } ( T )$ is the set of all isolated eigenvalues with finite multiplicity. H. Weyl proved (in a special case) the following result [[#References|[a4]]], which is now commonly known as Weyl's theorem: Let $T$ be self-adjoint and $S$ symmetric and compact. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013015.png" /></td> </tr></table>
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\begin{equation*} \sigma _ { \text{ess} } ( T ) = \sigma _ { \text{ess} } ( T + S ). \end{equation*}
  
In fact, it is even sufficient to require the relative compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013016.png" />, which means that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013017.png" /> is compact. Another variant of the Weyl theorem states that two self-adjoint operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013019.png" /> have the same essential spectrum if the difference of the resolvents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013020.png" /> is compact [[#References|[a3]]] (cf. also [[Resolvent|Resolvent]]). The Weyl theorem is very important in mathematical quantum mechanics, where it serves to prove that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013021.png" /> for a large class of two-body potentials. (A generalization to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120130/w12013023.png" />-body problem is the HVZ theorem by W. Hunziker, C. van Winter, and G. Zhislin, see [[#References|[a3]]].) The situation is considerably more complicated for non-self-adjoint operators, where there are several possible definitions of the essential spectrum [[#References|[a1]]]. In general, one defines the Weyl spectrum as the largest subset of the spectrum that is invariant under compact perturbations.
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In fact, it is even sufficient to require the relative compactness of $S$, which means that the operator $S ( T + i ) ^ { - 1 }$ is compact. Another variant of the Weyl theorem states that two self-adjoint operators $A$ and $B$ have the same essential spectrum if the difference of the resolvents $( A + i ) ^ { - 1 } - ( B + i ) ^ { - 1 }$ is compact [[#References|[a3]]] (cf. also [[Resolvent|Resolvent]]). The Weyl theorem is very important in mathematical quantum mechanics, where it serves to prove that $\sigma _ { ess } ( - \Delta + V ) = [ 0 , \infty )$ for a large class of two-body potentials. (A generalization to the $N$-body problem is the HVZ theorem by W. Hunziker, C. van Winter, and G. Zhislin, see [[#References|[a3]]].) The situation is considerably more complicated for non-self-adjoint operators, where there are several possible definitions of the essential spectrum [[#References|[a1]]]. In general, one defines the Weyl spectrum as the largest subset of the spectrum that is invariant under compact perturbations.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Gustafson,  "Necessary and sufficient conditions for Weyl's theorem"  ''Michigan Math. J.'' , '''19'''  (1972)  pp. 71–81</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Kato,  "Perturbation theory for linear operators" , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods in modern mathematical physics IV. Analysis of operators" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Weyl,  "Über quadratische Formen, deren Differenz vollstetig ist"  ''Rend. Circ. Mat. Palermo'' , '''27'''  (1909)  pp. 373–392</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  K. Gustafson,  "Necessary and sufficient conditions for Weyl's theorem"  ''Michigan Math. J.'' , '''19'''  (1972)  pp. 71–81</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  T. Kato,  "Perturbation theory for linear operators" , Springer  (1976)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Reed,  B. Simon,  "Methods in modern mathematical physics IV. Analysis of operators" , Acad. Press  (1978)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H. Weyl,  "Über quadratische Formen, deren Differenz vollstetig ist"  ''Rend. Circ. Mat. Palermo'' , '''27'''  (1909)  pp. 373–392</td></tr></table>

Latest revision as of 17:00, 1 July 2020

on operator perturbation

The spectrum $\sigma ( T )$ of a (possibly unbounded) self-adjoint operator $T$ is naturally divided into a point spectrum $\sigma _ { p } ( T )$ (i.e., the set of eigenvalues) and a continuous spectrum $\sigma _ { c } ( T )$, both subsets of the real axis (cf. also Spectrum of an operator). The continuous spectrum is very sensitive against perturbations. The Weyl–von Neumann theorem [a2], Sect. X.2.2, states that for any self-adjoint operator $T$ in a separable Hilbert space there exists a (compact) self-adjoint operator $S$ in the Hilbert–Schmidt class (cf. also Hilbert–Schmidt operator) such that $T + S$ has a pure point spectrum. Moreover, $S$ can be chosen arbitrarily small in the Hilbert–Schmidt norm. The set of eigenvalues obtained by perturbation of a continuous spectrum is dense in any interval covered by the continuous spectrum of the unperturbed operator. In spectral theory one introduces therefore another subset of $\sigma ( T )$ which is more stable under perturbations. The essential spectrum $\sigma _ { \text{ess} } ( T )$ is defined as $\sigma ( T ) \backslash \sigma _ { d } ( T )$, where $\sigma _ { d } ( T )$ is the set of all isolated eigenvalues with finite multiplicity. H. Weyl proved (in a special case) the following result [a4], which is now commonly known as Weyl's theorem: Let $T$ be self-adjoint and $S$ symmetric and compact. Then

\begin{equation*} \sigma _ { \text{ess} } ( T ) = \sigma _ { \text{ess} } ( T + S ). \end{equation*}

In fact, it is even sufficient to require the relative compactness of $S$, which means that the operator $S ( T + i ) ^ { - 1 }$ is compact. Another variant of the Weyl theorem states that two self-adjoint operators $A$ and $B$ have the same essential spectrum if the difference of the resolvents $( A + i ) ^ { - 1 } - ( B + i ) ^ { - 1 }$ is compact [a3] (cf. also Resolvent). The Weyl theorem is very important in mathematical quantum mechanics, where it serves to prove that $\sigma _ { ess } ( - \Delta + V ) = [ 0 , \infty )$ for a large class of two-body potentials. (A generalization to the $N$-body problem is the HVZ theorem by W. Hunziker, C. van Winter, and G. Zhislin, see [a3].) The situation is considerably more complicated for non-self-adjoint operators, where there are several possible definitions of the essential spectrum [a1]. In general, one defines the Weyl spectrum as the largest subset of the spectrum that is invariant under compact perturbations.

References

[a1] K. Gustafson, "Necessary and sufficient conditions for Weyl's theorem" Michigan Math. J. , 19 (1972) pp. 71–81
[a2] T. Kato, "Perturbation theory for linear operators" , Springer (1976)
[a3] M. Reed, B. Simon, "Methods in modern mathematical physics IV. Analysis of operators" , Acad. Press (1978)
[a4] H. Weyl, "Über quadratische Formen, deren Differenz vollstetig ist" Rend. Circ. Mat. Palermo , 27 (1909) pp. 373–392
How to Cite This Entry:
Weyl theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_theorem&oldid=16328
This article was adapted from an original article by Bernd Thaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article