Asymptotic Expansions for Characteristic Values and Functions of a Second Order Ordinary Linear Differential Operator

Citation

Swanson, Charles Andrew (1957) Asymptotic Expansions for Characteristic Values and Functions of a Second Order Ordinary Linear Differential Operator. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/NSHV-8P43. https://resolver.caltech.edu/CaltechETD:etd-08062004-144733

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Consider a second order ordinary linear differential operator on a real half-open interval (0,b] (b > 0) which contains no singular points. Suppose x = 0 is a singular point. The basic characteristic value problem is defined on this interval when suitable boundary conditions are adjoined at the endpoints. Two classes of perturbed characteristic value problems are defined on subintervals [a,b], where a is a small positive number. It is proved under certain conditions on the basic problem that for each isolated characteristic value [...] of the basic problem there is a characteristic value [...] of the perturbed problem which is developable in an asymptotic expansion with leading term [...] valid as [...]. Furthermore, the characteristic function corresponding to [...] possesses an asymptotic expansion valid as [...] uniformly in the interval [a,b]. These expansions are not asymptotic power series, but are asymptotic expansions of a more general type.

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