Joint Colloquium: Blower -- On tau functions arising from linear systems

This talk considers the Fredholm determinant $\det (I+\lambda \Gamma_{\phi_{(x)}})$ of a
Hankel integral operator on $L^2(0, \infty )$ with kernel $\phi (s+t+2x)$, where
$\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ is a matrix scattering function associated with a linear
system $(-A,B,C)$ on state space $H$.  The talk introduces an operator $R_x$ by
Lyapunov’s equation $dR_x/dx=-AR_x-R_xA$, then $\tau (x)=\det (I+R_x)$ satisfies $\tau
(x)=\det (I+\Gamma_{\phi_{(x)}})$.  There is an associated differential ring ${\bf S}$
of operators on the state space which gives a calculus extending P\"oppe’s results on
Hankel operators with scattering class symbols.  Also, $\tau$ generalizes the notion of
the $\theta$ function of an algebraic curve ${\cal E}$.  The tau functions are shown to
be consistent with isomonodromic tau functions in the sense of Jimbo, Miwa and Ueno, and
with Sato’s $\tau$ functions.  The talk discusses cases (i) and (ii).\par \indent (i)
$(2,2)$-admissible linear linear systems give scattering class potentials, with scalar
scattering function $\phi (x)=Ce^{-xA}B$.  Here a Gelfand--Levitan equation relates
$\phi$ and $u(x)$, which is solved with operators involving $R_x$.  Any linear system of
rational matrix ordinary differential equations gives rise to an integrable operator $K$
as in Tracy and Widom’s theory of matrix models.  Under general conditions on existence
of solutions, it is shown that there exist Hankel operators $\Gamma_\Phi$ and
$\Gamma_\Psi$ with matrix symbols such that $\det (I+\mu K)=\det (I+\mu
\Gamma_\Phi\Gamma_\Psi )$.  \par \indent (ii) The periodic linear system $\Sigma$ has a
tau function $\tau$ and a periodic potential $u$.  Hence $\Sigma$ is associated with
Hill’s discriminant $\Delta (\lambda )$ and a spectral curve ${\cal E}$, which is
typically a transcendental hyperelliptic curve of infinite genus.  The Jacobi variety
${\bf X}$ of ${\cal E}$ is then an infinite dimensional complex torus.  Periodic linear
systems give periodic potentials as in Hill’s equation $-\psi’’+u\psi =\lambda\psi$.  If
$u$ is real $C^2$ and periodic, and Hill’s equation has independent Floquet solutions
for all but finitely many $\lambda$, then $u$ is finite gap and $\tau$ is the
restriction of a theta function to a straight line in the Jacobian of a hyperelliptic
curve.  The talk deals with the case of elliptic $u$, so $u$ is expressed as a quotient
of tau functions from periodic linear systems.\par }