Weyl's theory and the method of complex rotation

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Abstract
The spectral properties of a singular second order differential operator L is analysed in terms of Weyl's theory. A spectral function is derived and the results are briefly compared with scattering theory. The analytic properties of L, its associated Green's function and related quantities of modern scattering theory are analysed. It is shown that the analytic continuation of an appropriately dilated operator L allows for an analytic extension of the singular case to a higher order Riemann sheet. The present theory is thus applicable to the study of the entire bound and continuous spectrum of L. A numerical realization of the theory is presented and applied to the scattering potential of Moiseyev et al. (1978, Molec. Phys., 36, 1613). New resonance structures are thereby found. Their asymptotic behaviour is discussed and an attempt to classify this type of structure is made. Implications of the present theory for predissociations in diatomic molecules, analysis of scattering experiments and ab initio studies are finally made.