Very little is known about the nature of the spectrum of 
for collision operators corresponding to real interactions.
The main features of the spectrum of 
, but not all of
its detailed properties are known for the hard sphere potential,
and for 
potentials [Grad
[1963],
Dorfman
[1963],
Kuscer and Williams
[1967], Yan and
Wannier
[1968] and Pao
[1974]]
.
The spectrum for Maxwell
molecules (
potential) is the only potential for which
the spectrum is known completely [see preceding refs]. Almost nothing is
known for
the potentials having an attractive component.
We have chosen a discrete spectrum because our exactly solvable
model has this structure, and we wish to understand the features
of this model in the first instance, and also because the
mathematics of discrete spectra is vastly simpler to that of
continuous spectra. (This is why the theory of compact
operators, and of bounded self-adjoint operators is so much more
developed than the case of general linear operators.) However,
the existence of runaway
[Howorka et al.
[1979], Moruzzi
and Kondo
[1980]]
shows that there are
circumstances
where this
assumption fails. In this phenomenon, there are regions of the
parameter 
for which the transport coefficients are not
well defined. The motion of the centroid is reminiscent of
acceleration rather that that of a steady drift velocity. The
arguments in the previous section rule out the possibility of
runaway arising when the spectrum is discrete. Consequently, we
can say that a necessary condition for runaway is that the
spectrum must have continuous or residual components.
Cavalleri and Paveri-Fontana [1972] give as
sufficient
condition for runaway to occur that the integral
of the velocity dependent
collision frequency should exist. This collision frequency is defined by
where 
is the background gas distribution function, and
is the total scattering cross section. Since
necessity and sufficiency are often closely connected
conditions in mathematics, one may speculate that there is a
strong connection between the asymptotic form of 
for
large c, and the structure of the spectrum of 
.