Type: Seminar

Distribution: World

Expiry: 12 Nov 2015

Auth: leo@dyn-129-78-250-115.wirelessguest.usyd.edu.au (ltzo2369) in SMS-WASM

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 }