- Dethankelshort
Submitted manuscript, 494 KB, PDF document

- Version 2 26/6/17
Submitted manuscript, 528 KB, PDF document

- https://arxiv.org/abs/1207.2143
Submitted manuscript

Research output: Working paper

Unpublished

Publication date | 5/04/2012 |
---|---|

Number of pages | 66 |

<mark>Original language</mark> | English |

\noindent {\bf Abstract} This paper considers the Fredholm determinant

$\det (I-\Gamma_x)$ of a Hankel integral operator on $L^2(0, \infty )$

with kernel $\phi (s+t+2x)$, where $\phi$ is a matrix scattering function.

The original contribution of the paper is a related operator $R_x$ such that $\det

(I-R_x)=\det (I-\Gamma_x)$ and $-dR_x/dx=AR_x+R_xA$ and an associated

differential ring. The paper introduces two main classes of linear systems $(-A,B,C)$ for Schr\"odinger's equation $-\psi''+u\psi =\lambda \psi$, namely\par

\indent (i) $(2,2)$-admissible linear linear systems which give

scattering class potentials, with scattering function $\phi (x)=Ce^{-xA}B$;\par

\indent (ii) periodic linear systems, which give periodic

potentials as in Hill's equation.\par

\indent The paper introduces the state ring ${\bf S}$ for linear systems as in (i) and (ii), and the tau function is $\tau (x) =\det (I+R_x)$. \par

\indent (i) A Gelfand--Levitan equation relates $\phi$ and

$u(x)=-2{{d^2}\over{dx^2}}\log\det (I-R_x)$, which is solved with linear systems as in inverse scattering. Any system of rational matrix differential equations

gives rise to an integrable operator $K$ as in Tracy and Widom's theory of matrix models. The Fredholm determinant $\det (I+\lambda K)$ equals $\det (I+\lambda \Gamma_\Phi\Gamma_\Psi )$, where $\Gamma_\Phi$ and $\Gamma_\Psi$ are Hankel operators with matrix symbols. The paper derives differential equations

for $\tau$ in terms of the singular points of the differential equation. This paper also introduces an admissible linear system with tau function which gives a solution of Painlev\'e's equation $P_{II}$.\par

\indent (ii) Consider Hill's equation with elliptic potential $u$. Then $u$ is expressed as a quotient of tau functions from periodic linear systems. If the general solution is a quotient of tau functions from periodic linear systems for all but finitely many complex eigenvalues, then $u$ is finite gap and has a hyperelliptic spectral curve.

The isospectral flows of

Schr\"odinger's equation are given by potentials $u(t,x)$ that evolve

according to the Korteweg de Vries equation $u_t+u_{xxx}-6uu_x=0.$

Every hyperelliptic curve ${\cal E}$ gives a solution for $KdV$ which

corresponds to rectilinear motion in the Jacobi variety of ${\cal E}$.

Extending P\"oppe's results, the paper develops a functional calculus for linear systems

thus producing solutions of the KdV equations. If $\Gamma_x$ has finite

rank, or if $A$ is invertible and $e^{-xA}$ is a uniformly continuous

periodic group, then the solutions are explicitly given in terms of

matrices.\par

$\det (I-\Gamma_x)$ of a Hankel integral operator on $L^2(0, \infty )$

with kernel $\phi (s+t+2x)$, where $\phi$ is a matrix scattering function.

The original contribution of the paper is a related operator $R_x$ such that $\det

(I-R_x)=\det (I-\Gamma_x)$ and $-dR_x/dx=AR_x+R_xA$ and an associated

differential ring. The paper introduces two main classes of linear systems $(-A,B,C)$ for Schr\"odinger's equation $-\psi''+u\psi =\lambda \psi$, namely\par

\indent (i) $(2,2)$-admissible linear linear systems which give

scattering class potentials, with scattering function $\phi (x)=Ce^{-xA}B$;\par

\indent (ii) periodic linear systems, which give periodic

potentials as in Hill's equation.\par

\indent The paper introduces the state ring ${\bf S}$ for linear systems as in (i) and (ii), and the tau function is $\tau (x) =\det (I+R_x)$. \par

\indent (i) A Gelfand--Levitan equation relates $\phi$ and

$u(x)=-2{{d^2}\over{dx^2}}\log\det (I-R_x)$, which is solved with linear systems as in inverse scattering. Any system of rational matrix differential equations

gives rise to an integrable operator $K$ as in Tracy and Widom's theory of matrix models. The Fredholm determinant $\det (I+\lambda K)$ equals $\det (I+\lambda \Gamma_\Phi\Gamma_\Psi )$, where $\Gamma_\Phi$ and $\Gamma_\Psi$ are Hankel operators with matrix symbols. The paper derives differential equations

for $\tau$ in terms of the singular points of the differential equation. This paper also introduces an admissible linear system with tau function which gives a solution of Painlev\'e's equation $P_{II}$.\par

\indent (ii) Consider Hill's equation with elliptic potential $u$. Then $u$ is expressed as a quotient of tau functions from periodic linear systems. If the general solution is a quotient of tau functions from periodic linear systems for all but finitely many complex eigenvalues, then $u$ is finite gap and has a hyperelliptic spectral curve.

The isospectral flows of

Schr\"odinger's equation are given by potentials $u(t,x)$ that evolve

according to the Korteweg de Vries equation $u_t+u_{xxx}-6uu_x=0.$

Every hyperelliptic curve ${\cal E}$ gives a solution for $KdV$ which

corresponds to rectilinear motion in the Jacobi variety of ${\cal E}$.

Extending P\"oppe's results, the paper develops a functional calculus for linear systems

thus producing solutions of the KdV equations. If $\Gamma_x$ has finite

rank, or if $A$ is invertible and $e^{-xA}$ is a uniformly continuous

periodic group, then the solutions are explicitly given in terms of

matrices.\par

Due to be submitted for publication in refereed mathematical journal.