Direct and inverse scattering for selfadjoint Hamiltonian systems on the line.(English)Zbl 0970.34072

An inverse scattering theory is constructed for the selfadjoint system of differential equations $-iJ_{2n}Y'(x,\lambda)-V(x)Y(x,\lambda)=\lambda Y(x,\lambda),\quad -\infty<x<\infty,$ on the line, with $J_{2n}=\left[\begin{matrix} I_n & 0 \\ 0 & -I_n\end{matrix}\right],\qquad V(x)=\left[\begin{matrix} 0 & k(x) \\ k^*(x) & 0 \end{matrix}\right],$ $$I_n$$ is the identity matrix of odd order $$n$$, $$k(x)\in L(-\infty,\infty)$$ is the $$n\times n$$-matrix function, and $$*$$ denotes the matrix conjugate transpose.

MSC:

 34L25 Scattering theory, inverse scattering involving ordinary differential operators 81U40 Inverse scattering problems in quantum theory 34A55 Inverse problems involving ordinary differential equations 74J25 Inverse problems for waves in solid mechanics 47A40 Scattering theory of linear operators
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