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A relation connecting certain magnitudes which characterize the scattering of particles with magnitudes characterizing their absorption. More exactly, the dispersion relation is a relation connecting the real part of the scattering amplitude (in the more general case, the [[Green function|Green function]]) with certain types of integrals of its imaginary part. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334001.png" /> be absolutely integrable on the axis, and let it satisfy the causal relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334003.png" />. Then its Fourier–Laplace transform
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<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/d/d033/d033400/d0334004.png" /></td> </tr></table>
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will be a holomorphic function in the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334005.png" />, and the real and imaginary parts of the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334006.png" /> will satisfy the dispersion relation
+
A relation connecting certain magnitudes which characterize the scattering of particles with magnitudes characterizing their absorption. More exactly, the dispersion relation is a relation connecting the real part of the scattering amplitude (in the more general case, the [[Green function|Green function]]) with certain types of integrals of its imaginary part. Let a function  $  f ( t) $
 +
be absolutely integrable on the axis, and let it satisfy the causal relation $  f ( t) = 0 $,
 +
$  t < 0 $.
 +
Then its Fourier–Laplace transform
  
<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/d/d033/d033400/d0334007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
\widetilde{f}  ( \zeta )  = \int\limits f ( t) e ^ {i \zeta t }  dt ,
 +
\  \zeta = p + iq ,
 +
$$
  
In describing real physical processes the dispersion relation of the type (*) becomes more complicated, since the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334008.png" /> may increase at infinity as a polynomial (in this case a dispersion relation with subtractions is obtained), the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d0334009.png" /> may be a generalized function of slow growth, while the number of variables may be more than one (multi-dimensional dispersion relations).
+
will be a holomorphic function in the upper half-plane  $  q > 0 $,
 +
and the real and imaginary parts of the boundary value  $  \widetilde{f}  ( p) $
 +
will satisfy the dispersion relation
 +
 
 +
$$ \tag{* }
 +
\mathop{\rm Re}  \widetilde{f}  ( p)  = 
 +
\frac{1} \pi
 +
v _ {p} \int\limits _ {- \infty } ^  \infty 
 +
 
 +
\frac{ \mathop{\rm Im}  \widetilde{f}  ( p  ^  \prime  )  d p  ^  \prime  }{p  ^  \prime  - p }
 +
.
 +
$$
 +
 
 +
In describing real physical processes the dispersion relation of the type (*) becomes more complicated, since the function $  \widetilde{f}  ( \zeta ) $
 +
may increase at infinity as a polynomial (in this case a dispersion relation with subtractions is obtained), the boundary value $  \widetilde{f}  ( p) $
 +
may be a generalized function of slow growth, while the number of variables may be more than one (multi-dimensional dispersion relations).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Bogolyubov,  B.V. Medvedev,  M.K. Polivanov,  "Questions in the theory of dispersion relations" , Moscow  (1958)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,  A.A. Logunov,  A.I. Oksak,  I.T. Todorov,  "General principles of quantum field theory" , Kluwer  (1990)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Bogolyubov,  B.V. Medvedev,  M.K. Polivanov,  "Questions in the theory of dispersion relations" , Moscow  (1958)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.N. Bogolyubov,  A.A. Logunov,  A.I. Oksak,  I.T. Todorov,  "General principles of quantum field theory" , Kluwer  (1990)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A dispersion relation of the type defined here is often called a Kramers–Kronig relation. In the classical dispersion of light the relation gives a connection between the real (dispersive) and imaginary (absorptive) parts of the index of refraction.
 
A dispersion relation of the type defined here is often called a Kramers–Kronig relation. In the classical dispersion of light the relation gives a connection between the real (dispersive) and imaginary (absorptive) parts of the index of refraction.
  
Consider a linear wave equation such as the beam equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d03340010.png" />. For a sinusoidal wave train <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d03340011.png" /> to satisfy such an equation some relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d03340012.png" /> between the frequency and the wave number must hold. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d03340013.png" />. This relation is called the dispersion relation. There are generalizations to non-linear wave equations, e.g., the KdV-equation, where the dispersion relation also involves the amplitude. Dispersion relations for waves are extensively discussed in [[#References|[a5]]].
+
Consider a linear wave equation such as the beam equation $  \phi _ {tt} + \gamma  ^ {2} \phi _ {xxxx} = 0 $.  
 +
For a sinusoidal wave train $  \phi ( x , t ) = A  \mathop{\rm exp} ( i k \cdot x - i \omega t ) $
 +
to satisfy such an equation some relation $  G ( k , \omega ) = 0 $
 +
between the frequency and the wave number must hold. In this case $  \omega  ^ {2} - \gamma  ^ {2} k  ^ {4} = 0 $.  
 +
This relation is called the dispersion relation. There are generalizations to non-linear wave equations, e.g., the KdV-equation, where the dispersion relation also involves the amplitude. Dispersion relations for waves are extensively discussed in [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Kronig,  ''J Opt. Soc. Amer'' , '''12'''  (1926)  pp. 547</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.A. Kramers,  , ''Atti. Congr. Intern. Fisici Como'' , '''2'''  (1927)  pp. 545</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.G. van Kampen,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d03340014.png" />-matrix and causality condition I. Maxwell field"  ''Phys. Rev.'' , '''89'''  (1953)  pp. 1072–1079</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.G. van Kampen,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033400/d03340015.png" />-matrix and causality condition II. Nonrelativistic particles"  ''Phys. Rev.'' , '''91'''  (1953)  pp. 1267–1276</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Bremermann,  "Distributions, complex variables, and Fourier transforms" , Addison-Wesley  (1965)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G.B. Whitham,  "Linear and non-linear waves" , Wiley  (1974)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Kronig,  ''J Opt. Soc. Amer'' , '''12'''  (1926)  pp. 547</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.A. Kramers,  , ''Atti. Congr. Intern. Fisici Como'' , '''2'''  (1927)  pp. 545</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.G. van Kampen,  "S-matrix and causality condition I. Maxwell field"  ''Phys. Rev.'' , '''89'''  (1953)  pp. 1072–1079</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.G. van Kampen,  "S-matrix and causality condition II. Nonrelativistic particles"  ''Phys. Rev.'' , '''91'''  (1953)  pp. 1267–1276</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Bremermann,  "Distributions, complex variables, and Fourier transforms" , Addison-Wesley  (1965)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G.B. Whitham,  "Linear and non-linear waves" , Wiley  (1974)</TD></TR></table>

Latest revision as of 11:56, 26 March 2023


A relation connecting certain magnitudes which characterize the scattering of particles with magnitudes characterizing their absorption. More exactly, the dispersion relation is a relation connecting the real part of the scattering amplitude (in the more general case, the Green function) with certain types of integrals of its imaginary part. Let a function $ f ( t) $ be absolutely integrable on the axis, and let it satisfy the causal relation $ f ( t) = 0 $, $ t < 0 $. Then its Fourier–Laplace transform

$$ \widetilde{f} ( \zeta ) = \int\limits f ( t) e ^ {i \zeta t } dt , \ \zeta = p + iq , $$

will be a holomorphic function in the upper half-plane $ q > 0 $, and the real and imaginary parts of the boundary value $ \widetilde{f} ( p) $ will satisfy the dispersion relation

$$ \tag{* } \mathop{\rm Re} \widetilde{f} ( p) = \frac{1} \pi v _ {p} \int\limits _ {- \infty } ^ \infty \frac{ \mathop{\rm Im} \widetilde{f} ( p ^ \prime ) d p ^ \prime }{p ^ \prime - p } . $$

In describing real physical processes the dispersion relation of the type (*) becomes more complicated, since the function $ \widetilde{f} ( \zeta ) $ may increase at infinity as a polynomial (in this case a dispersion relation with subtractions is obtained), the boundary value $ \widetilde{f} ( p) $ may be a generalized function of slow growth, while the number of variables may be more than one (multi-dimensional dispersion relations).

References

[1] N.N. Bogolyubov, B.V. Medvedev, M.K. Polivanov, "Questions in the theory of dispersion relations" , Moscow (1958) (In Russian)
[2] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[3] N.N. Bogolyubov, A.A. Logunov, A.I. Oksak, I.T. Todorov, "General principles of quantum field theory" , Kluwer (1990) (Translated from Russian)

Comments

A dispersion relation of the type defined here is often called a Kramers–Kronig relation. In the classical dispersion of light the relation gives a connection between the real (dispersive) and imaginary (absorptive) parts of the index of refraction.

Consider a linear wave equation such as the beam equation $ \phi _ {tt} + \gamma ^ {2} \phi _ {xxxx} = 0 $. For a sinusoidal wave train $ \phi ( x , t ) = A \mathop{\rm exp} ( i k \cdot x - i \omega t ) $ to satisfy such an equation some relation $ G ( k , \omega ) = 0 $ between the frequency and the wave number must hold. In this case $ \omega ^ {2} - \gamma ^ {2} k ^ {4} = 0 $. This relation is called the dispersion relation. There are generalizations to non-linear wave equations, e.g., the KdV-equation, where the dispersion relation also involves the amplitude. Dispersion relations for waves are extensively discussed in [a5].

References

[a1] R. Kronig, J Opt. Soc. Amer , 12 (1926) pp. 547
[a2] H.A. Kramers, , Atti. Congr. Intern. Fisici Como , 2 (1927) pp. 545
[a3] N.G. van Kampen, "S-matrix and causality condition I. Maxwell field" Phys. Rev. , 89 (1953) pp. 1072–1079
[a4] N.G. van Kampen, "S-matrix and causality condition II. Nonrelativistic particles" Phys. Rev. , 91 (1953) pp. 1267–1276
[a5] H. Bremermann, "Distributions, complex variables, and Fourier transforms" , Addison-Wesley (1965)
[a6] G.B. Whitham, "Linear and non-linear waves" , Wiley (1974)
How to Cite This Entry:
Dispersion relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dispersion_relation&oldid=12664
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article