Asymptotics of the m-Coefficient for Eigenvalue Problems with Eigenparameter in the Boundary Conditions

For Sturm-Liouville problems on a, ) with a regular -dependentboundary condition at a, and the limit point case at , a techniqueof W. N. Everitt 1] is employed to obtain asymptotic formulaefor the associated m()-functions on rays and lines in the complex-plane. The method relies on asymptotic formulae for solutionsof the initial value problem for –u'+qu = u, as || ,which the author has given in 4]. For the case of the regularleft endpoint, the asymptotic formulae on vertical lines sufficeto provide a direct proof of the formula for the total variationof the associated spectral function, a question which the authorhad raised in 3; Remark 5.2].