On tau functions associated with linear systems

TL;DR

This paper explores tau functions linked to linear systems, introducing operator functions satisfying Lyapunov's equation, and establishes conditions for integrability of Schrödinger equations with meromorphic potentials, with applications to integrable PDEs.

Contribution

It introduces a new operator function R_x satisfying Lyapunov's equation, extending tau function theory beyond self-adjoint cases, and develops a differential ring framework for analyzing integrability.

Findings

Tau functions are expressed as determinants of operators satisfying Lyapunov's equation.

Conditions are derived for Schrödinger equations with meromorphic potentials to be integrable.

Constructs solutions to KP equations and verifies Fay's identities.

Abstract

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\bf C}$ and state space $H$. The function $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$ on $L^2((0, \infty ); {\bf C})$; 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 ${\cal A}$ of operators in $H$ and a differential ring homomorphism $\lfloor\,\,\rfloor :{\cal A}\rightarrow {\bf C}[u,u', \dots ]$ such that $u=-4\lfloor A\rfloor$, which provides a substitute for the multiplication rules for Hankel operators considered by P\"oppe, and McKean {\sl Cent. Eur. J. Math.} {\bf 9} (2011), 205--243. The paper obtains conditions on $(-A,B,C)$ for Schr\"odinger's equation with meromorphic $u$ to be integrable by quadratures. Special results apply to the linear systems associated with scattering $u$, periodic $u$ and elliptic $u$. The paper constructs a family of solutions to the Kadomtsev--Petviashivili differential equations, and proves that certain families of tau functions satisfy Fay's identities.\par