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Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review
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TY - GEN
T1 - Tau functions for linear systems
AU - Blower, Gordon
AU - Newsham, Samantha
PY - 2021/3/6
Y1 - 2021/3/6
N2 - 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
AB - 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
KW - integrable systems
KW - Fredholm determinant
KW - inverse scattering
M3 - Conference contribution/Paper
T3 - International Workshop on Operator Theory and Applications
BT - Operator theory advances and applications
A2 - Bastos, Amelia
A2 - Castro, Luis
A2 - Karlovich, Alexei
PB - Springer Birkhäuser
ER -