On tau functions associated with linear systems
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 Schrödinger'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 selfadjoint scattering case {\sl Commun. Math. Phys.} {\bf 47} (1976), 171183. This paper introduces a operator function $R_x$ that satisfies Lyapunov's equation ${{dR_x}\over{dx}}=AR_xR_xA$ and $\tau (x)=\det (I+R_x)$, without assumptions of selfadjointness. When $A$ is sectorial, and $B,C$ are HilbertSchmidt, there exists a noncommutative 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 Pöppe, and McKean {\sl Cent. Eur. J. Math.} {\bf 9} (2011), 205243. The paper obtains conditions on $(A,B,C)$ for Schrödinger'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 KadomtsevPetviashivili differential equations, and proves that certain families of tau functions satisfy Fay's identities.\par
 Publication:

arXiv eprints
 Pub Date:
 July 2012
 DOI:
 10.48550/arXiv.1207.2143
 arXiv:
 arXiv:1207.2143
 Bibcode:
 2012arXiv1207.2143B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 47B35;
 34B25
 EPrint:
 This paper has been rewritten and the current version replaces the first version on ArXiv