Asymptotics of the m -Coefficient for Eigenvalue Problems with Eigenparameter in the Boundary Conditions
Charles T. Fulton · Bulletin of the London Mathematical Society · 1981
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].