The main objective of this paper is to extend the pioneering work of Sims in [9] on secondorder linear differential equations with a complex coefficient, in which he obtains an analogue of the Titchmarsh-Weyl theory and classification. The generalisation considered exposes interesting features not visible in the special case in [9]. An m-function is constructed (which is either unique or a point on a "limit-circle") and the relationship between its properties and the spectrum of underlying maccretive differential operators analysed. The paper is a contribution to the study of non-self-adjoint operators; in general the spectral theory of such operators is rather fragmentary, and further study is being driven by important physical applications, to hydrodynamics, electro-magnetic theory and nuclear physics, for instance.

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1. IntroductionIn a recent paper [3], an extended Liouville–Green formulaformula herewas developed for solutions of the second-order differential equationformula hereHere γM(x) ∼Q−¼(x), QM(x) ∼Q−½(x) and εM(x)→0 as x→∞, while M([ges ]2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M−1. The general form of (1·1) had been obtained previously by Cassell [5], [6], [7] and Eastham [10], [11, section 2·4]. In particular, the proof of (1·1) in [10] and [11] depended on the formulation of (1·2) as a first-order system and then on a process of M repeated diagonalization of the coefficient matrices in a sequence of related differential systems.

We consider the spectral function ρ(µ) (µ ≥ 0) for the Sturm-Liouville equation y ′′ + (λ − q)y = 0 on [0, ∞) with the boundary condition y(0) = 0 and where q has slow decay O(x −α ) (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333-348) to deal with the new theoretical and computational complexities which arise for slow decay.