On tau functions associated with linear systems

This paper 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 $\phi$ is a matrix scattering function associated with a linear system $(-A,B,C)$. The original contribution of the paper is a related operator $R_x$ such that $\det (I-R_x)=\det (I-\Gamma_x)$ and $-dR_x/dx=AR_x+R_xA$ and an associated differential ring ${\bf S}$ of operators on the state space. The paper introduces two main classes of linear systems $(-A,B,C)$ for Schr\"odinger's equation $-\psi"+u\psi =\lambda \psi$, namely (i) $(2,2)$-admissible linear linear systems which give scattering class potentials, with scattering function $\phi (x)=Ce^{-xA}B$; (ii) periodic linear systems, which give periodic potentials as in Hill's equation. The paper analyses ${\bf S}$ for linear systems as in (i) and (ii), and the tau function is $\tau (x) =\det (I+R_x)$. The isospectral flows of Schr\"odinger's equation are given by potentials $u(t,x)$ that evolve according to the Korteweg de Vries equation $u_t+u_{xxx}-6uu_x=0.$ Every hyperelliptic curve ${\cal E}$ gives a solution for $KdV$ which corresponds to rectilinear motion in the Jacobi variety of ${\cal E}$. Extending P\"oppe's results, the paper develops a functional calculus for linear systems, thus producing solutions of the KdV equations. If $\Gamma_x$ has finite rank, or if $A$ is invertible and $e^{-xA}$ is a uniformly continuous periodic group, then the solutions are explicitly given in terms of matrices.