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Tau functions for linear systems

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSNConference contribution/Paperpeer-review

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Tau functions for linear systems. / Blower, Gordon; Newsham, Samantha.
Operator theory advances and applications: IWOTA Lisbon 2019. ed. / Amelia Bastos; Luis Castro; Alexei Karlovich. Springer Birkhäuser, 2021. (International Workshop on Operator Theory and Applications).

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSNConference contribution/Paperpeer-review

Harvard

Blower, G & Newsham, S 2021, Tau functions for linear systems. in A Bastos, L Castro & A Karlovich (eds), Operator theory advances and applications: IWOTA Lisbon 2019. International Workshop on Operator Theory and Applications, Springer Birkhäuser. <https://www.springer.com/gp/book/9783030519445>

APA

Blower, G., & Newsham, S. (in press). Tau functions for linear systems. In A. Bastos, L. Castro, & A. Karlovich (Eds.), Operator theory advances and applications: IWOTA Lisbon 2019 (International Workshop on Operator Theory and Applications). Springer Birkhäuser. https://www.springer.com/gp/book/9783030519445

Vancouver

Blower G, Newsham S. Tau functions for linear systems. In Bastos A, Castro L, Karlovich A, editors, Operator theory advances and applications: IWOTA Lisbon 2019. Springer Birkhäuser. 2021. (International Workshop on Operator Theory and Applications).

Author

Blower, Gordon ; Newsham, Samantha. / Tau functions for linear systems. Operator theory advances and applications: IWOTA Lisbon 2019. editor / Amelia Bastos ; Luis Castro ; Alexei Karlovich. Springer Birkhäuser, 2021. (International Workshop on Operator Theory and Applications).

Bibtex

@inproceedings{67eaeb98f26549bfb7e726899d346887,
title = "Tau functions for linear systems",
abstract = "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",
keywords = "integrable systems, Fredholm determinant, inverse scattering",
author = "Gordon Blower and Samantha Newsham",
year = "2021",
month = mar,
day = "6",
language = "English",
series = "International Workshop on Operator Theory and Applications",
publisher = "Springer Birkh{\"a}user",
editor = "Amelia Bastos and Luis Castro and Alexei Karlovich",
booktitle = "Operator theory advances and applications",

}

RIS

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 -