演講公告 90年1月9日 (星期二)

摘要

Abstract
We consider the problem of determining a funcdamental matrix $\psi$
$$d\psi/dx = aJ\psi + Q(x)\psi, \; det \psi(x, z) \ne 0, \tag 1$$
here $J = diag(d_1, d_2, ..., d_n), d_j$ are distinct. We may look for $\psi$ of the form $\psi = m \exp(xzJ)$. We consider the following equation for $m$
$$dm/dx = z[J, m] + Qm, \; z \notin \sum \tag 2$$
$m(\cdot, z)$ bounded and absolutely continuous $m(x, z) \to I$ as $x \to -\infty$, here $\sum = \{z: Re(zd_j) = Re(zd_k)$, some $j \ne k\}$. We may convert (2) into a Fredholm integral equation.
The first column $m_1$ of $m$ satisfies a Volterra equation. But the remaining columns $m_j$ of $m$ satisfy Fredholm equations. Following the idea of Beals-Coiman, the wedge products $m_1 \land m_2, m_1 \land m_2 \land m_3, ..., m_1 \land m_2 \land m_3 \cdot\cdot\cdot m_n$ satisfy Volterra equation . If we normalize $m$ at $x = \infty$, then the wedge products $m_{n-1}\land m_n, ... m_1 \land m_2 \land \cdot\cdot\cdot \land m_n$ also satisfy Volterra equations. A satisfactory scattering transform is defined. The associated inverse problem is also solved.