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Explicit Wiener-Hopf factorization for certain non-rational matrix functions. (English) Zbl 0790.47012

The present paper introduces a new class of \(2 \times 2\) nonrational matrix functions for which explicit Wiener-Hopf factorizations may be constructed. The functions are of the form \[ G(k,x)=\left( {T(k) \atop - L(k)e^{-2ikx}} {-R(k)e^{2ikx} \atop T(k)} \right), \] where \(x\) is a real parameter and factorization is with respect to the variable \(k\). The scalar functions \(T(\cdot)\), \(R(\cdot)\) and \(L(\cdot)\) are assumed to be meromorphic on the upper half plane \(\mathbb{C}^ +\) with continuous boundary values on the extended real line, \(T(\infty)=1\), the values \(R(k)\) and \(L(k)\) vanish as \(k \to \infty\) in \(\overline {\mathbb{C}^ +}\), either \(T(0) \neq 0\) or \(T(k)\) vanishes linearly at \(k=0\), and \(T(k) \neq 0\) for \(k \in \overline {\mathbb{C}^ +} \backslash \{0\}\). Furthermore, for any real \(x\) \[ G(k,x)^{-1}={0\;1 \choose 1\;0}G(-k,x){0\;1 \choose 1\;0},\quad k \in \mathbb{R}, \] and \(G(k,x)\), as a function of \(k\in\mathbb{R}\), belongs to a suitable Banach algebra of \(2\times 2\) matrix functions within which Wiener-Hopf factorization is possible (e.g., the Wiener algebra). The problem to construct Wiener-Hopf factorizations for \(2\times 2\) matrix functions of the above type appears in inverse scattering problems for the 1-D Schrödinger equation and some related Schrödinger-type equations. In this case, \(T(k)\) is the transmission coefficient, and \(R(k)\) and \(L(k)\) are the reflection coefficients from the right and the left, respectively. The Wiener-Hopf factors of the matrix \(G(k,x)\) are obtained explicitly by the contour integration method. When \(T(k)\) has a zero at \(k=0\), the factorization becomes noncanonical. The connections with the Riemann-Hilbert barrier problem version of the inverse scattering problem are also described.

MSC:

47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
81U40 Inverse scattering problems in quantum theory
47A40 Scattering theory of linear operators
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