Method of complex scaling

Incidentally we note that in addition to the methods pursued in this monograph, i.e., the exact solution of Schrodinger s equation, there are other methods available for determining resonance energies and widths, namely the method of complex scaling (Reinhardt 1982 Junker 1982 Ho 1983 Moiseyev 1984) and the stabilization procedure (Taylor 1970 Meier, Cederbaum, and Domcke 1980 Lefebvre 1985). 

In some previous papers [3], the author has ivestigated the properties of pairs of adjoint operators T and T+ in general, and the present paper may be considered as an extension of these studies. In the Uppsala group, there has also been a considerable interest in the properties of non-self-adjoint operators in connection with the method of complex scaling [4] with results which may be considered as illustrations of the more general theorems treated here. 

On the Change of Spectra Associated with Unbounded Similarity Transformations of a Many-Particle Hamiltonian and the Occurrence of Resonance States in the Method of Complex Scaling. Part I. General Theory... 

In the physical applications, one is starting from wave functions ifr = tj/(x) which are originally defined only on the real axis, i.e., for — oo < x < +oo, and, in order to use the method of complex scaling, it is then necessary to continue these functions analytically from the real axis out in the complex plane. Some of these problems have been studied in greater detail by mathematicians (2), but they are also of essential interest to theoretical physicists. 

In this section we will consider the method of complex scaling (2) as a typical example of an unbounded similarity transformation of the restricted type. It is here sufficient to consider a single one-dimensional particle with the real coordinate x( — oo < x < +qo), since the IV-particle operator U in a 3N-dimensional system may then be built up by using the product constructions given by Eqs. (2.23) and (2.25). 

The Editor would like to thank the authors for their contributions, which give an interesting picture of the current range from studies of the use of the Lie algebra in quantum theory, overestimates of the neutrino mass, to relativistic effects in atoms and molecules and the calculation of lifetimes of metastable states by means of the method of complex scaling. 

In conclusion, the method of complex scaling as an unbounded similarity transformation of the restricted type is briefly discussed, and some numerical applications containing complex eigenvalues - which may be related to resonance states... 

This study was started in order to find out whether one could find meaningful complex eigenvalues in the Hartree-Fock scheme for a transformed Hamiltonian in the method of complex scaling. This problem was intensely discussed at the 1981 Tarfala Workshop in the Kebnekaise area of the Swedish mountains. It was found that, if the many-electron Hamiltonian undergoes a similarity transformation U which is a product of one-electron transformations u - as in the method of complex scaling - then the Fock-Dirac operator p as well as the effective Hamiltonian Heff undergo one-electron similarity... 

The puzzle depended on the simple fact that most physicists using the method of complex scaling had not realized that the associated operator u - the so-called dilatation operator - was an unbounded operator, and that the change of spectra -e.g. the occurrence of complex eigenvalues - was due to a change of the boundary conditions. Some of these features have been clarified in reference A, and in this paper we will discuss how these properties will influence the Hartree-Fock scheme. The existence of the numerical examples finally convinced us that the Hartree-Fock scheme in the complex symmetric case would not automatically reduce to the ordinary Hartree-Fock scheme in the case when the many-electron Hamiltonian became real and self-adjoint. Some aspects of this problem have been briefly discussed at the 1987 Sanibel Symposium, and a preliminary report has been given in a paper4 which will be referred to as reference D. 

Quite a few years have now gone since the Tarfala Workshop in 1981, and many new results have appeared in the method of complex scaling. The question of the occurrence of complex eigenvalues corresponding to so-called resonance states in the Hartree-Fock approximation for various types of many-electron Hamiltonians is, however, still not completely solved, and the authors feel that time is now mature to bring up these problems for more intense discussions. 

Here f= f(z) is an analytic function originally defined on the real axis x, which is assumed to be analytic also in the point z = t x. An essential problem in the method of complex scaling in quantum mechanics is hence to study whether a wave function y = y(x) defined on the real axis may be continued analytically out in the complex plane to the point z = rjx. Since many analytic functions have natural boundaries, it is from the very beginning evident that there may be considerable restrictions on the parameter q itself. More generally, one may define the operator u through the relation... 

The above-mentioned three-step transformation procedure allows us to expand the scope of the method of complex scaling to virtually any system. The method has been applied successfully to the study of predissociating resonances of several vdW molecules. Including Ar-H, Ar-HD, Ar-N and Ar-HCl etc.. Involving either piece-wise or numerical potentials. 

A particular class of 5 states with both electrons located on the same side of the nucleus, considered by Richter and Wintgen (113), was also investigated by Prunele (137) by the method of o(4,2) operator replacements generalized by the method of complex scaling, which gave rather qualitative results in comparison with the very accurate results of Richter and Wintgen. 

---