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