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For Sturm‐Liouville problems on [a, ∞) with a regular λ‐dependent boundary condition at a, and the limit point case at ∞, a technique of W. N. Everitt [1] is employed to obtain asymptotic formulae for the associated m(λ)‐functions on rays and lines in the complex λ‐plane. The method relies on asymptotic formulae for solutions of the initial value problem for −u″+qu = λu, as |λ| → ∞, which the author has given in [4]. For the case of the regular left endpoint, the asymptotic formulae on vertical lines suffice to provide a direct proof of the formula for the total variation of the associated spectral function, a question which the author had raised in [3; Remark 5.2].
Bulletin of the London Mathematical Society – Wiley
Published: Nov 1, 1981
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