Title:

Linear systems and determinants in integrable systems

The thesis concerns linear systems and scattering theory. In particular, it presents
lineal' systems for some integrable systems and finds discrete analogues for many well known
results for continuous variables. It introduces some new tools from linear systems
and applies them to standard integrable systems. We begin by expressing the first Painleve
equation as the compatibility condition of a certain Lax pair and introduce the Kortewegde
Vries partial differential equation. We introduce the spectral curve for algebraic families
and the Toda lattice. The Fredholm determinant of a trace class Hankel integral operator
gives rise to a tau function. Dyson used the tau function to solve an inverse spectral
problem for Schrodinger operators. When a plane wave is subject to Schrodinger's equation
and scattered by a potential u, the output is described at great distances by a scattering
function. The spectral problem is to find the spectrum of Schrodinger's operator in L2
and hence the scattering function. The inverse spectral problem is to find the potential
given the scattering function. The scattering and inverse scattering problems are linked by
the Gelfand Levitan equation. In this thesis, for a discrete linear system, we introduce a
scattering function and Hankel matrix and a version of the GelfandLevitan equation for
discrete linear systems. We introduce the discrete operator ∑∞/k=n AkBCAk and use
it to solve the GelfandLevitan equation and compute Fredholm determinants of Hankel
operators. We produce a discrete analogue of a calculation of Poppe giving a solution to
the Kortewegde Vries equation and via the methods of linear systems find an analogous
solution in terms of Hankel matrices. We then produce a discrete analogue of the Miura
transform. Thus the main new contributions of this thesis are the discrete analogues of the R
operator, the Gelfand Levitan equation, the Lyapunov equation and the Miura transform.
