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Finite Frequency Kramers Kronig Relations, or

Checking the compatibility of experimental measurements
of the complex dielectric constant

Dispersion relations are prevalent throughout physics
and derive from the causal nature of the response of
materials, bodies or particles to electromagnetic, elastic or other fields.
The classic example of a dispersion relation is the Kramers-Kronig (KK)
relation (Kramers, H.A., Nature, 117 (1926) 775,
Kronig, R. de L., J. Opt. Soc. Am. 12 (1926) 547)
that couples the real and imaginary
parts of the complex dielectric constant of a material by

where *P* denotes the principle value of the integral, where
the last identity follows from the fact that
. Thus is the Hilbert transform of . The chief obstacle to the practical application of the KK relation
is that one needs to know over all frequencies
to determine , whereas a given experiment
only yields values of and
over a finite frequency range.
This is typically handled in a crude manner which is only sometimes effective:
some approximation for is used outside the measured
frequency range.
When the measured and computed function disagree,
one is left in doubt as to whether the measurements of and are compatible with each other, or whether the data set violates what
is known about the analytic properties of including the positivity of for . Our work (Milton, G.W., Eyre, D.J. and Mantese, J.V *
Finite Frequency Range Kramers Kronig Relations:
Bounds on the Dispersion,* Phys. Rev. Lett. 79 (1997) 3062-3064)
(request a reprint)
addressed this by deriving sharp bounds on the function
for given at *m* frequencies between and and
given over the same frequency interval.
The appearance of bounds is natural and
reflects the incompleteness of our knowledge of .
We now propose to address the question
of what bounds can be placed on given the value
of over two disjoint frequency intervals.
In other words we want to derive bounds which test the compatibility of
data taken over two disjoint frequency ranges. This is
particularly useful when a different experimental apparatus is used to
take the measurement over the second frequency range. Mathematically
the problem reduces to obtaining sharp bounds which correlate the
values a Stieltjes function can take at *m*+1 points in the complex
plane given that the measure associated with the Stieltjes function
is supported on three disjoint known intervals on the real axis. This is quite
similar to the problem solved in our paper which involved
Stieltjes functions supported on two disjoint known intervals.
The generalization should be straightforward.

Analogous KK relations exist which give the real part of the complex
magnetic permeability and the complex bulk and shear moduli
in terms of their positive imaginary parts. Therefore our bounds should
apply equally well to testing the compatibility
of experimental data for these magnetic and viscoelastic moduli.
With appropriate normalization (to capture the correct
high frequency limit) they should also apply to the frequency
dependent response of electrical networks and elastic
structures. In addition they should be applicable (with minor modification) in particle
physics (Taylor, J. R. *Scattering theory: the quantum
theory on nonrelativistic collisions*, Wiley, New York, NY, 1972)
specifically to testing the compatibility of
measurements of the complex forward scattering amplitude
collected over a finite range of energies.

*David Eyre*

*2/27/1998*
This was supported by the General Motors Corporation and
the National Science Foundation through grants DMS-9803748,
DMS-9629692 and DMS-9402763.