## 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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### References:

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