On reducing the canonical system to two dual differential systems (Q5934250)

Spectral problems for the matrix differential system NEWLINE\[NEWLINE -\frac{d}{dx}\left( P(x)\frac{du_1}{dx}\right) = z^2P(x)u_1 NEWLINE\]NEWLINE are solved by transferring to the canonical system NEWLINE\[NEWLINE \frac{dU}{dx} = izJH(x)U(x,z), NEWLINE\]NEWLINE with \(U(x,z) = (u_1(x,z), u_2(x,z))^{\top}\), where \(u_1(x,z), u_2(x,z)\) are \(m\times m\)-matrices, NEWLINE\[NEWLINE J= \begin{pmatrix} 0 & I_m \\ I_m & 0 \end{pmatrix}, NEWLINE\]NEWLINE and \(H(x)= \text{diag}( P(x),P^{-1}(x))\), with \(P(x)\) a continuous positively definite \(m\times m\)-matrix function. So, the matrix case (\(m\geq 1\)) can be studied, and a simpler procedure for finding \(P(x)\) by the known spectral data is obtained.