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On the spectrum of second-order differential operators with complex coefficients

B. M. Brown, D. K. R. McCormack, W. D. Evans, M. Plum

Abstract

The main objective of this paper is to extend the pioneering work of Sims on second-order linear differential equations with a complex coefficient, in which he obtains an analogue of the Titchmarsh–Weyl theory and classification. The generalization considered exposes interesting features not visible in the special case in Sims paper from 1957. 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 m-accretive 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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