View source for 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 <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

<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>

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

<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>

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).

====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>



====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 <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]]].

====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>