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- W1488474063 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 ϕ (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 Schrödinger’s equation − f″ + uf = λf, and derived the formula for the potential $$u(x)=-2{{d^2}over {dx^2}}log tau (x)$$ in the self-adjoint scattering case (Commun Math Phys 47:171–183, 1976). 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 $$lfloor ,,rfloor :{mathcal A}to mathbb {C}[u,u', dots ]$$ such that u = −4⌊A⌋, which extends the multiplication rules for Hankel operators considered by Pöppe and McKean (Cent Eur J Math 9:205–243, 2011)." @default.
- W1488474063 created "2016-06-24" @default.
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- W1488474063 date "2021-01-01" @default.
- W1488474063 modified "2023-09-26" @default.
- W1488474063 title "Tau Functions Associated with Linear Systems" @default.
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- W1488474063 doi "https://doi.org/10.1007/978-3-030-51945-2_5" @default.
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