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Rigorous asymptotic approximations of the WKB (or LiouvilleGreen) type are obtained for a basis of solutions to y′′ + [α + g(t)]y = 0 in the framework of C∗-algebras. Both cases α > 0 and α < 0 are included, thus generalizing the classical theory for scalar equations developed by F.W.J. Olver to matrix as well as to infinite-dimensional equations.

WKB-type approximations for second-order differential equations in $C*$-algebras