Abstract
A deep theorem of de Branges establishes the uniqueness of real normalized solutions of the inverse monodromy problem for 2 x 2 canonical systems of differential equations with a sym-plectic monodromy matrix which is inner with respect to the signature matrix \( \left[ {\begin{array}{*{20}{c}}0&i \\{ - i}&0\end{array}} \right] \) In this paper, this theorem is used to show that the inverse monodromy problem for such sys ems of equations has a unique normalized solution for monodromy matrices which are J-inner with respect to any 2 x 2 signature matrix which is not definite if and only if the monodromy matrix U (λ) has zero exponential type in either the upper or lower half plane (or equivalently, if and only if the exponential type of U(λ) is equal to the exponential type of its determinant). A complete description of the set of solutions is furnished in the opposite case. Enroute simple proofs are given for a number of well known type formulas.
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Arov, D.Z., Dym, H. (2001). Some Remarks on the Inverse Monodromy Problem for 2 x 2 Canonical Differential Systems. In: Bart, H., Ran, A.C.M., Gohberg, I. (eds) Operator Theory and Analysis. Operator Theory: Advances and Applications, vol 122. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8283-5_3
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DOI: https://doi.org/10.1007/978-3-0348-8283-5_3
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