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Tau functions for linear systems

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Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space $\mathbb{C}$ and state space $H$. The function $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$ on $L^2((0, \infty ); \mathbb{C})$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+ \Gamma_{\phi_{(x)}})$ defines the tau function of $(-A,B,C)$. Such tau functions arise in Tracy and Widom's theory of matrix models, where they describe the fundamental probability distributions of random matrix theory. Dyson considered such tau functions in the inverse spectral problem for Schr\"odinger's equation $-f''+uf=\lambda f$, and derived the formula for the potential $u(x)=-2{{d^2}\over{dx^2}}\log \tau (x)$ in the self-adjoint scattering case {\sl Commun. Math. Phys.} {\bf 47} (1976), 171--183. This paper introduces a operator function $R_x$ that satisfies Lyapunov's equation ${{dR_x}\over{dx}}=-AR_x-R_xA$ and $\tau (x)=\det (I+R_x)$, without assumptions of self-adjointness. When $-A$ is sectorial, and $B,C$ are Hilbert--Schmidt, there exists a non-commutative differential ring ${\mathcal A}$ of operators in $H$ and a differential ring homomorphism $\lfloor\,\,\rfloor :{\mathcal A}\rightarrow \mathbb{C}[u,u', \dots ]$ such that $u=-4\lfloor A\rfloor$, which extends the multiplication rules for Hankel operators considered by P\"oppe, and McKean {\sl Cent. Eur. J. Math.} {\bf 9} (2011), 205--243. \par