One of the origins of the challenging problems and simultaniously new ideas in modern analysis and mathematical physics is the theory of exactly solvable quantum field and statistical physics models. Perhaps the most intriguing among these problems, is the problem of the calculation of the relevant correlation functions.
For $1+1$ exactly solvable quantum models their correlation functions
can be represented as Fredholm determinants of the integral operators whose
kernels, $K(\lambda_{1}, \lambda_{2 })$, have the following special form :
(K*)
\begin{equation}
K(\lambda_{1}, \lambda_{2}) =
\frac{\sum_{j=1}^{N}f_{j}(\lambda_{1})g_{j}(\lambda_{2})}
{\lambda_{1}-\lambda_{2}},\qquad
\sum_{j} f_{j}(\lambda)g_{j}(\lambda)=0,
\end{equation}
(*)
where functions $f_{j}(\lambda), g_{j}(\lambda)$, and, in fact, the
contour of integration, depend on the model under consideration.
The first representation of such type was obtained in 1966 by Lenard
for equal-time correlation functions in one-dimensional
impenetrable bosons. Later on, determinant formulae were derived for
a majority of excatly solvable statistical mechanics and quantum
field models (works of Izergin, Korepin, McCoy, Perk, Shrock, Slavnov,
Tracy, and others).
The integral operators (K*) possess some remarkable properties, which are discussed extensively in the literature (works of Harnad, Its, Izergin, Korepin, Slavnov, Tracy, Widom). One of these properties is that the resolvent kernel corresponding to kernel (K*) can be explicitly evaluated in terms of the solution of the matrix Riemann-Hilbert problem with the jump matrix $G(\lambda)$ given by the equation:
\begin{equation}\label{rh} G_{jk}(\lambda)=\delta_{jk} - 2\pi i f_{j}(\lambda)g_{k}(\lambda). \end{equation}
This observation was made in 1990 by Its, Izergin, Korepin, and Slavnov, and it generalizes some of the technical ideas of the well known paper of Jimbo, Miwa, Mori, and Sato (1980). The indicated property of kernels (K*) brings the Riemann-Hilbert method of the theory of integrable nonliner PDEs into the theory of exactly solvable quantum and statistical mechanics models. It allows, in particular, to develop a new powerful approach to the asymptotic evaluation of the correlation functions. The approach was originated in the end of the eighties in the works of Its, Izergin, Korepin and Slavnov, and has been further developed in the nineties in the works of Deift, Essler, Frahm, Its, Izergin, Korepin, Novokshenov, Slavnov, Varzugin, Waldron, and Zhou. The main analytic tool of the modern version of the scheme is the nonlinear steepest descent method for oscillatory Riemann-Hilbert problems suggested in 1993 by Deift and Zhou.
Tentative Course Outline:
Lecture 1.
Integrable Fredholm Kernels and the Riemann-Hilbert Problem. Elements
of general theory. Concrete examples. Riemann-Hilbert problem and the
integrable differential equations for correlation functions.
Lecture 2.
Asymptotic analysis of oscillatory Riemann-Hilbert problems arising in
the theory of integrable systems. Nonlinear steepest descent method.
Lecture 3.
Free fermion models. Large time and distance asymptotics of correlation
functions of impenetrable bosons and XX0 antiferromagnet.
Lecture 4.
Nonfree fermion models. Operator valued Riemann-Hilbert problems.
One dimensional bosons at finite coupling. XXX and XXZ models. Further
perspectives.