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A complex scale transformation of the time-independent Schrodinger equation leads to a symmetric eigenvalue problem containing both bound states and resonance (complex) eigenvalues as solutions. An extended virial theorem is stated, and its necessary fulfiHment is pointed out. The latter, in conjunction with a symmetric stationary principle, allows for determination of resonance (complex) eigenvalues by means of elementary matrix manipulations. Application to the Stark effect in the hydrogen atom shows agreement with previous calculations based on numerical integration. t I. INTRODUCTION.Continuum orbitals and wave functions are needed in order to, describe and analyze radiationless processes. The calculation of transition energies and moments between discrete states are well-known procedures, whether one uses a conventional wave-' function picture or the recent powerful propagator technique. ' When the continuum is involved, the ubiquity of nonquantization leads to a situation of a more difficult nature. It is our aim, however, to show in this paper that a simpIe extension of known stationary principles allows for a direct determination of resonance (complex) eigenvalues by means of standard matrix manipulations.

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Weyl's theory for a singular second-order differential equation and the complex scaling method of Balslev and . Combes are combined to obtain a stable method for describing the continuous spectrum. The method obtained can be viewed as an extension of the Siegert method. The theory is applied to a model potential earlier used by Moiseyev et al.The experimental development of laser spectroscopy, electron scattering, and in particular, the high-frequency deflection technique, ' have lead to a considerable interest in the theory of the spectral continuum and more specifically in quasibound states and resonances in the vibrational continua of molecules. The simultaneous development of complex scaling in terms of the Aguilar-Balslev-Combes (ABC) theory' ' for dilatation analytic operators has opened a field of new ideas. Methods of this type are important for the understanding of a complete, nonisolated system. One of the first applications of the ABC theory in the study of resonances was reported by Bain, Bardsley, Junker, and Sukumar. ' They used a modified variational principle to obtain the complex eigenvalue of the complex-rotated Hamilton-ian. The position of the resonance as a function of the dilatation angle was studied numerically. One of the difficulties of the application of this method is its basis-set dependence. This problem was to some extent solved utilizing the existence of the complex virial theorem Weyl's theory for a singular second-order differential equation' has earlier proven to be an efficient tool in the analysis of the continuous spectrum. ' In the numerical applications made so far (see, e.g. , Hehenberger et al. ' and Ref. 8) one makes use of the numerical information of the Green's function or the Weyl-Titchmarsh m function on the real axis, A Siegert state" can then only be obtained via analytic continuation based on the previously mentioned numerical data.In this report we present a synthesis of Weyl's theory and the theory of complex scaling. The step to "go into" the complex plane appears, in fact, quite natural if the details of Weyl's theory are considered. In contrast to previous techniques which one way or another were based on the numerical dependence of the resonance so-y' (x)+~2[a -q(x)]y(x} = 0.(l)For simplicity we will only consider one equation here. Extensions to the case of coupled equations are easily incorporated in the present formulation, but this will be reported elsewhere. Putting~gg, where g is a complex scale factor with 0~argy & v/4, we obtain the transformed equation, 2 2 y'"(x) +~" [e(q)q"(x)]y"(x) = O.In order to emphasize the g dependence, we will write Eq. (2) as a differential equation with the real variable x belonging to the interval ( -, +~) and the parametric dependence on g indicated by a suffix. We assume that g above is consistent with (2) limq"(x}= q"(+~) = const&~. lutions on the dilatation angle 8 = argy, we employ a direct numerical integration of the Siegert' solution on the higher-order Riemann sheet. As a consequence, the dependence of the...

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Simon's exterior-scaling procedure is applied to model systems within the framework of the finite-basis-set approximation. We show that if the basis-set functions are scaled when r )ro, a variational solution is obtained by adding the term , 5(r -ro)[d/drk][lexp( -2iB)] to the Hamiltonian, where k is the logarithmic derivative of the resonance wave function at ro. Two computational methods are proposed. One is an ab initio linear variational iterative procedure, whereas in the second one A, is semiclassically estimated during the iteration procedure and all the variational calculations are carried out for r ro and therefore are exactly 6 independent. An illustrative numerical application is presented.