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Series representation and asymptotics for Titchmarsh-Weyl m-functions. (English) Zbl 0715.34044

Summary: Sturm-Liouville and Dirac operators are considered on \(a\leq x<\infty\), and it is assumed that the coefficients are integrable on [a,\(\infty)\). The Titchmarsh-Weyl m-function is given as the ratio of two series which converge on Im \(\lambda\geq 0\) for \(| \lambda |\) sufficiently large. For coefficients where N-th derivatives exist and are integrable on [a,\(\infty)\), an asymptotic expansion to order \(\lambda^{-N}\) is derived which is valid on Im \(\lambda\geq 0\) as \(\lambda\to \infty\). Proofs are given in detail only for the Dirac system. These results contrast with previous work in that the representations are valid for real as well as complex \(\lambda\).

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B24 Sturm-Liouville theory
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